# Dynamics of shock waves in a superfluid unitary Fermi gas

Wen Wen<EMAIL_ADDRESS>Department of Mathematics and Physics, Hohai
University, Changzhou 213022, China  Tiankun Shui  Yafei Shan  College of
Mechanical and Electronic Engineering, Hohai University, Changzhou 213022,
China  Changping Zhu  Jiangsu Key Laboratory of Power Transmission and
Distribution Equipment Technology, Changzhou Key Laboratory of Sensor Networks
and Environmental Sensing, Hohai University, Changzhou 213022, China

###### Abstract

We study the formation and dynamics of shock waves initiated by a repulsive
potential in a superfluid unitary Fermi gas by using the order-parameter
equation. In the theoretical framework, the regularization process of shock
waves mediated by the quantum pressure term is purely dispersive. Our results
show good agreement with the experiment of Joseph et al. [Phys. Rev. Lett.
106, 150401 (2011)]. We reveal that the boxlike-shaped density peak observed
in the experiment consists of many vortex rings due to the transverse
instability of the dispersive shock wave. In addition, we study the transition
from a sound wave to subsonic shock waves by increasing the strength of the
repulsive potential and show a strong qualitative change in the propagation
speed of the wavefronts. In the relatively small strength regime, the speed
decreases below the sound speed with increasing the strength as a scaling
behavior, while in the large regime the speed remains almost unchanged, which
is found to be the same expansion speed of the proliferation of the vortex
rings.

###### pacs:

03.75.Kk, 03.75.Lm, 03.75.Ss

##  I Introduction

The unitary Fermi gas is a system of fermions with an infinite two-body
scattering length, which has remarkable universal properties and has
connections with areas as diverse as nuclear physics and high-$T_{c}$
superconductivity zwe2012 . The unprecedented experimental sophistication
reached recently in the unitary Fermi gas has opened a unique opportunity for
systematic studies of elementary excitations in a strongly interacting Fermi
system ket2008 ; all2012 . Experiments have observed collective modes kin2004
; bar2004 ; tey2013  and sound waves jos2007 ; sid2013 , nonlinear topological
structures including vortices zwi2005  and dark solitons yef2013 ; ku2014 ,
and another specific nonlinear phenomenon, i.e. shock waves jos2011 . Shock
wave generated in a classical fluid is characterized by a well-defined shock
front across which, there is a dissipation from the effect of viscosity to
avoid the onset of a gradient catastrophe whi1974 ; lan1987 . In contrast to
viscous shock wave, shock wave in a Bose-Einstein condensate is mediated by
dispersion. Such dispersive shock wave associated with a fast oscillating
wave-train trailing has been predicted theoretically kul2003 ; dam2004 ;
kam2004 ; vic2004 ; hof2006 , and then observed in various experiments cha2008
; dut2001 ; mep2009 .

In the recent experiment performed by the Thomas’s group jos2011 , the
formation and propagation of shock waves in a unitary Fermi gas were studied
by colliding between two clouds. They reproduced the experimental data well by
hydrodynamic equations with a phenomenological viscosity term. On the other
hand, Bulgac et al. bul2012  and Salasnich et al. anc2012 ; sal2011 ; anc2013
; lsa2013  independently argued that the observed fermionic shock front is
dominated not by dissipation but rather dispersion, because at zero
temperature the bulk viscosity vanishes son2007 ; cao2011  and the shear
viscosity is at a minimum cao2011 ; wal2012 . Since the structures of shock
wave fronts are too fine to be imaged in the typical Fermi gas experiments,
the regularization mechanism is difficult to be directly revealed through
density measurements. In order to determine the appropriate regularization
mechanism, Lowman and Hoefer low2013  subsequently proposed to measure the
propagating speed of shock waves which is regularization dependence hof2006 .
It is noticed that the above mentioned references anc2012 ; sal2011 ; anc2013
; lsa2013 ; low2013  mainly present the results of one dimensional (1D)
dynamics of fermonic shock waves by integrating over the transverse
coordinate. There is a lack of discussion on the transversal effect on shock
wave dynamics, however, which is always present in Fermi gas experiments and
of considerable practical interest.

In this work, we use the order-parameter equation to perform a detailed three-
dimensional (3D) numerical simulation of all stages in the experiment of
fermionic shock waves jos2011  and compare to experiment with good agreement.
We find that a dispersive shock wave forms, which possess an expanding
oscillatory wavetrain with a large amplitude. Due to the transverse
instability, the wavetrain decays into a large number of vortex rings in a
short time. We interpret that the observed boxlike-shaped density peak in the
experiment actually originates from the proliferation of the vortex rings.
Furthermore, we study the mechanism of transition from sound wave propagation
to shock wave dynamics and present how the wavefront speed depends on the
strength of the repulsive potential which creates it. For the smallest
strength creating a linear wave, the propagation speed is in good agreement
with the theoretically predicted sound speed. As increasing the strength in
the relatively weak regime, we find that the speed decreases below the sound
speed. In this regime we derive an analytical expression for the speed as a
function of the strength from an effective 1D model, which is also confirmed
by the full 3D numerical simulation exactly. As the strength increases towards
the moderate value, the transversal dynamics takes an effect on the properties
of shock waves, and the speed deviates the scaling behavior predicted by the
1D model. As increasing the strength in the sufficiently large regime, shock
waves are formed by colliding two spatially separated clouds, and the
propagation speeds of the wavefronts are found to be unchanged. We understand
the speed independent on the strength as the expansion speed of the vortex
rings, which are originated from the decay of the dispersive shock waves due
to the transverse instability.

This paper is organized as follows: In Sec. II, we solve numerically the
order-parameter equation to study the formation and dynamics of fermionic
shock waves in the realistic system. In Sec. III, we analytically and
numerically study the transition from sound wave propagation to subsonic shock
wave dynamics by calculating the propagation speeds of the wavefronts. Last
section gives a conclusion.

##  II Comparison with experiment

###  II.1 Theoretical description

We consider an ultracold Fermi gas at zero temperature, in which fermionic
atoms have two spin states with equal number and all atoms are paired in the
superfluid state. The dynamic behaviors of Fermi superfluids can be described
by the following time dependent order-parameter equation sal2008 ; sal12008 ;
adh2008 ; adh12008 ; wen2008

$i\hbar\frac{\partial\Psi_{s}}{\partial
t}=\left[-\frac{\hbar^{2}\nabla^{2}}{4m}+2V_{\rm ext}({\bf
r})+2\mu(n)\right]\Psi_{s},$ (1)

where $\Psi_{s}$ is the order parameter of fermionic atomic pairs in the
condensed state and the atomic density is given by $n=2|\Psi_{s}|^{2}$, $m$ is
the mass of atom, and $V_{{\rm ext}}({\bf r})$ is the external potential. In
the unitary limit the s-wave scattering length becomes infinity and no
characteristic length is set, so the equation of state has the form ast2004 ;
bul2006

$\mu(n)=\xi\frac{\hbar^{2}}{2m}(3\pi^{2})^{2/3}n^{2/3},$ (2)

where the Bertsch parameter $\xi=0.38$ is tuned to fit the experimental
results nav2010 ; ku2012 .

This complex order parameter can be specified by
$\Psi_{s}=\sqrt{\frac{n}{2}}e^{i\Phi_{s}}$, where $\Phi_{s}$ is the phase of
fermionic pair condensates. By introducing the superfluid velocity given by
the gradient of the phase ${\bf v}=(\hbar/2m)\nabla\Phi_{s}$, from the order-
parameter equation one can deduce the quantum hydrodynamics equations

$\displaystyle\frac{\partial n}{\partial t}+\nabla\cdot(n{\bf v})=0,$ (3a)
$\displaystyle m\frac{\partial{\bf v}}{\partial
t}+\nabla\left[\frac{1}{2}mv^{2}+\mu(n)+V_{{\rm ext}}({\bf
r})-\frac{1}{4}\frac{\hbar^{2}}{2m}\frac{\nabla^{2}\sqrt{n}}{\sqrt{n}}\right]=0,$
(3b)

which includes a gradient correction of von-Weizs$\ddot{a}$cker energy form
zub2009 ; zub12009 ; and2010  (i.e. quantum pressure term)

$\lambda\frac{\hbar^{2}}{2m}\frac{\nabla^{2}\sqrt{n}}{\sqrt{n}}$ (4)

with $\lambda=1/4$. In absence of the quantum pressure term, Eqs.(3) reduce to
the classical hydrodynamic equations lan1987 ; men2002 .

In order to prevent the density gradient catastrophe and model the shock wave
formation, Ref. jos2011  introduces a dissipative viscosity term
$\nu\partial_{z}(n\partial_{z}v)/n$ into the normalized 1D form of the
classical hydrodynamic equations, where $\nu=10\hbar/m$ is fitted to the
experimental results. However, it is shown that Fermi gases in the unitary
regime have a vanishing viscosity at zero temperature son2007 ; cao2011 ;
wal2012 . For the quantum hydrodynamics equations (3), an alternative
regularization mechanism can be naturally proposed, without the
phenomenological viscosity term. The gradient term (4) that takes into account
corrections to the kinetic energy due to spatial variations in the density of
the system, plays a role of a pure dispersive effect for the formation of
shock waves. There has been a lot of discussion about the value of the
coefficient $\lambda$ and2010 ; kim2004 ; sal12008 . The quantum pressure term
depending explicitly on the reduced Planck constant $\hbar$ gives quantization
condition, and the value of $\lambda$ has profound consequences on dynamics of
superfluid unitary Fermi gases anc2012 ; anc2013 ; lsa2013 .

Our calculations are based on the order-parameter equation, which models only
the dynamics of superfluid components. To further include the mechanism for
superfluid relaxation, such as pair-breaking, superfluid-normal transition,
and various photon processes, one must resort to time-dependent density
functional theory (DFT) which provides a microscopic description bulg2013 ;
bulg2011 . Recently based on DFT, many different dynamical processes have been
investigated, including formations of vortex lattices by stirring bulg2011 ,
crossing and reconnection of two vortex lines bulg2011 , dynamics of shock
waves bul2012 , oscillations of a vortex ring bul2014 , and conversion between
vortex rings and vortex lines due to a breaking of the axial symmetry of traps
wla2014 , some of which can not be obtained from the order-parameter equation.
While the order-parameter equation fails to model the pair-breaking mechanism
and normal components, the order-parameter equation depending on a single
collective wave function is significantly easier to solve analytically and
numerically than DFT, which requires to solve hundreds of thousands of
wavefunctions by means of supercomputing resources. In addition, Forbes and
Sharma forb2013  have presented a comparison between the dynamics of
superfluid unitary Fermi gases using DFT and the order-parameter equation
(they call the extended Thomas-Fermi model), and demonstrated that the order-
parameter equation is a good description for low-frequency dynamics of unitary
Fermi gases.

###  II.2 Numerical results

Figure 1: (Color online) Integrated atomic density profiles (in units of
$10^{-3}/\mu m^{2}$ per particle) after expansion for 1.5 ms after evolving in
the trap for different times: from left to right $t=$ 0 ms, 2 ms, 4 ms, 6 ms,
8 ms, and 10 ms. Here $x$ and $z$ correspond to the coordinates in $\mu m$,
with $x\in(-100,100)$ and $z\in(-210,210)$.

We solve the order-parameter equation numerically adh12010  to reproduce as
closely as possible the experimental condition. The experimental procedure
including three steps is described as follows jos2011  : (1) The unitary Fermi
gas containing a total of $N=2\times 10^{5}$ atoms is initially confined in a
cigar-shaped harmonic trap $V_{\rm ho}({\bf
r})=\frac{1}{2}m[\omega^{2}_{\perp}(x^{2}+y^{2})+\omega^{2}_{z}z^{2}]$ with
$\omega_{\perp}=2\pi\times 437\;{\rm Hz}$ and $\omega_{z}=2\pi\times
27.7\;{\rm Hz}$, and bisected by a repulsive potential $V_{\rm
rep}(z)=V_{0}\exp{[-(z-z_{0})^{2}/\sigma_{z}^{2}]}$ with strength
$V_{0}=12.7\;\mu K$, width $\sigma_{z}=21.2\;{\rm\mu m}$ and offset
$z_{0}=-5\;{\rm\mu m}$. (2) The repulsive potential $V_{\rm rep}$ is then
suddenly turned off, allowing for the two separated parts of the cloud to
collide with each other in the harmonic trap for a given hold time $t$. (3)
The absorption images are finally taken after an additional $1.5\;{\rm ms}$
expansion, during which the harmonic potential in the $r$-direction is
extinguished and the $z$-direction frequency is changed to
$\omega_{z}=2\pi\times 20.4\;{\rm Hz}$.

The creation of initial density perturbations undergoing shock wave dynamics
achieved experimentally can be classified into two ways dam2004 ; cha2008 ;
sal2011 ; mep2009 : suddenly turning off a repulsive potential produces a dip
in the density, which splits into two negative perturbations propagating in
opposite directions; alternatively, raising suddenly a repulsive potential
results in a density bump, which splits into two positive perturbations. Due
to the density-dependent speed that is regions of high density move with a
faster local velocity than regions of low density, as they travel the
steepenings of leading edges of positive density waves and of trailing edges
of negative ones are apparent. Therefore, according to the experimental
condition, two negative wave packets propagating in opposite directions are
actually observed.

Figure 2: (Color online) Closeup of the atomic density slices $n(x,0,z)$ (left
panels) and the corresponding phase profiles $\Phi_{s}(x,0,z)$ (right panels)
after expansion for 1.5 ms after evolving in the trap for different times,
from top left to bottom right: $t=0.7\;{\rm ms}$, $1.1\;{\rm ms}$, $1.2\;{\rm
ms}$, $1.3\;{\rm ms}$, $1.5\;{\rm ms}$ and $2.0\;{\rm ms}$. The color
corresponds to the phase: $\Phi_{s}=0$ through $2\pi$ is represented by the
sequence blue-green-red-blue.

In Figure 1, we show the numerical results for column atomic densities
$\int\\!dy\;n(x,y,z,t)$ by integrating along the $y$ axis at different hold
times $t$. It is seen that the presence of the repulsive potential leads to
two clearly separated parts. After the potential is rapidly turned off, the
two clouds expand and collide at the center of the trap, and a pronounced
bulge of higher atom density forms. As the density bulge spreads out from the
center of the trap, its front (back) edge self-steepens, which can be
identified as the formation of a shock wave. Notice that the observed front
and back edges actually correspond to the trailing edges of two negative waves
propagating in opposite directions, respectively. The result on the feature of
the box-like structure is found to be in good agreement with the experimental
observation [see Fig. 1 of Ref. jos2011 ]. However, clear soliton trains in
the wake of the density bulge which are indication of the dispersive mechanism
are not been found, as that in the experiment.

To better visualize the complicated density distributions at the center of the
system and find the dispersive effect, in Figure 2 we present zoomed-in views
for the atomic density slices by the plane $y=0$, that is $n(x,y=0,z,t)$, and
the corresponding phase profiles $\Phi_{s}(x,0,z)$. A closer look to the
center region reveals that a soliton train forms after evolving in the trap at
$t=0.7$ ms. Subsequently the soliton train is curved as it develops. The
curving structure is not symmetric about the $z=0$ axis, due to the offset
$z_{0}=-5\;{\rm\mu m}$ of the initial repulsive potential. When $t=1.3$ ms,
the transverse instability causes the quick decay of the soliton train into
the nucleation of vortex rings bul2014 ; cet2013 ; mun2014 ; hoe2009 ; wen2013
; fed2000 , evidenced by the phase profiles. After 2.0 ms, a large number of
vortex rings spread over the entire extent of the system. The formation of
vortex rings in an initial short time is shown to be difficult to detect
unambiguously in the experiment that are integrated along the line of sight
jos2011 ; cha2008 . While the phenomena are similar to those found in atomic
condensates cha2008 , they are first revealed in a superfluid unitary Fermi
gas. Recently based on DFT bul2012 , Bulgac et al. studied the shock wave
formed in the unitary Fermi gases trapped by a two-dimensional harmonic
potential. Because the particle number of the calculated system is too small,
no transverse instability is found and the dynamics is characterized by the
formation of a soliton train that eventually fills the entire superfluid. Our
numerical simulation can be consistent with this (not shown here) if we use
small number particle.

##  III transition from a sound wave to subsonic shock waves

Now we study the mechanism of the transition from sound wave propagation to
shock wave dynamics by increasing the strength of the repulsive potential and
calculate the propagation speeds of the wavefronts. Before presenting the
results from a full 3D numerical calculation, we reduce Eqs. (3) into an
effective 1D model, derive an analytical solution for the propagating speed
that is only valid for small perturbations, and make a comparison with the
numerical results.

###  III.1 Analytical derivation

We consider a cylindrically symmetric external potential composed of the
harmonic potential in the transverse direction and an axial potential:
$V_{{\rm ext}}({\bf r},t)=\frac{1}{2}m\omega^{2}_{\perp}r^{2}+V_{1}(z,t)$,
with $r=\sqrt{x^{2}+y^{2}}$. The atomic density and superfluid velocity are
then expressed as low2013

$n({\bf r},t)=n_{\perp}[r;n_{1}(z,t)]n_{1}(z,t)\;\;\;\;{\rm and}\;\;\;\;{\bf
v}({\bf r},t)=v_{1}(z,t).$ (5)

The atomic density satisfies the normalization conditions: $2\pi\int
dr\>rn_{\perp}=1$ and $\int\>dzn_{1}=N$. We assume sufficiently tight
transverse confinement so that its dynamics is neglected. Under the Thomas-
Fermi (TF) approximation, the density distribution in the equilibrium is given
by

$n_{\perp}[r;n_{1}]=n_{\perp}(0)(1-\frac{r^{2}}{R_{\perp}^{2}})^{3/2},\;\;\;\;\;R^{2}_{\perp}=\frac{2\mu_{\perp}}{m\omega^{2}_{\perp}}.$
(6)

By the normalization condition $2\pi\int^{R_{\perp}}_{0}drrn_{\perp}$=1, the
center atomic density and chemical potential are then given by

$n_{\perp}(0)=(\frac{2m\mu_{\perp}}{\xi\hbar^{2}})^{3/2}(6\pi^{2}n_{1})^{-1},\;\;\;\;\mu_{\perp}=\bar{\mu}n^{2/5}_{1}=(\frac{15\pi
m\omega^{2}_{\perp}}{4})^{2/5}(\frac{\xi\hbar^{2}}{2m})^{3/5}n^{2/5}_{1},$ (7)

respectively. Substituting the above ansatz into Eqs. (3) and integrating over
the transverse coordinate, one can arrive at the effective 1D hydrodynamic
equations

$\displaystyle\frac{\partial n_{1}}{\partial t}+\frac{\partial}{\partial
z}(n_{1}v_{1})=0,$ (8a) $\displaystyle m\frac{\partial{v_{z}}}{\partial
t}+\frac{\partial}{\partial
z}\left[\frac{1}{2}mv_{z}^{2}+\bar{\mu}n^{\frac{2}{5}}_{1}+V_{1}\right]=0,$
(8b)

with $\bar{\mu}$ defined in Eq. (7). Notice that the power of the nonlinear
term for the density has changed from $2/3$ to $2/5$ as we pass from the 3D
decription Eqs. (3) to the effective 1D model Eqs. (8). In additon, we assume
that the spatial scale of the density perturbation is larger than the healing
length and neglect the gradient correction (i.e. quantum pressure) term.

In the case of the axial harmonic potential
$V_{1}=\frac{1}{2}m\omega^{2}_{z}z^{2}$, one has the ground-state solution of
Eq. (8b)

$n_{1}(z)=n_{1}(0)(1-\frac{z^{2}}{R^{2}_{z}})^{5/2},\;\;\;\;R^{2}_{z}=\frac{2\mu_{G}}{m\omega^{2}_{z}},$
(9)

where $\mu_{G}$ is the ground-state chemical potential fixed by the normalized
condition $\int^{R_{z}}_{-R_{z}}dzn_{1}=N$. It is easy to get the explicit
expressions

$n_{1}(0)=(\mu_{G}/\bar{\mu})^{5/2},\;\;\;\;\;\mu_{G}=\sqrt{\xi}E_{F},$ (10)

with the Fermi energy $E_{F}=\hbar(3N\omega^{2}_{\perp}\omega_{z})^{1/3}$.
After linearization of Eqs. (8), one can obtain the local speed of sound
calculated at the center of the trap cap2006 ; wen2010

$c^{2}_{s}=\frac{2\bar{\mu}}{5m}n^{2/5}_{1}(0)=\frac{1}{5}\xi^{\frac{1}{2}}v^{2}_{F}$
(11)

with the Fermi velocity $v_{F}=\sqrt{2E_{F}/m}$ . We find that such local
sound speed $c_{s}=\sqrt{\xi^{1/2}/5}v_{F}$ is smaller than the sound speed
$c_{\rm ho}=\sqrt{\xi/3}v_{F}$ of a homogeneous system hei2006  by a factor of
$\sqrt{3/5\xi^{1/2}}=0.987$, which is resulted from the suppression of the
transverse confinement.

We perform the ansatz on the density $n_{1}(z,t)=n_{1}(0)\rho(z,t)$, which
implies a homogenous density background characterized by the central density.
By substituting it and Eq. (11) into Eqs. (8), the effective 1D hydrodynamic
equations are written as

$\displaystyle\frac{\partial\rho}{\partial t}+\rho\frac{\partial
v_{z}}{\partial z}+v_{z}\frac{\partial\rho}{\partial z}=0,$ (12a)
$\displaystyle\frac{\partial{v_{z}}}{\partial t}+v_{z}\frac{\partial
v_{z}}{\partial z}+\frac{c^{2}_{ls}}{\rho}\frac{\partial\rho}{\partial z}=0,$
(12b)

with

$c_{ls}(\rho)=c_{s}\rho^{\frac{1}{5}}.$ (13)

In order to find wave solutions of Eqs. (12), it is assumed that the velocity
$v_{z}$ depends explicitly the density $\rho$ sal2011 . In this case one can
put $\partial v_{z}/\partial
t=\frac{dv_{z}}{d\rho}\frac{\partial\rho}{\partial t}$, $\partial
v_{z}/\partial z=\frac{dv_{z}}{d\rho}\frac{\partial\rho}{\partial z}$, and
reduce Eqs. (12) to a hyperbolic equation

$\frac{\partial\rho}{\partial t}+C(\rho)\frac{\partial\rho}{\partial z}=0.$
(14)

It is easy to find the exact traveling wave solution of the hyperbolic
equation: $\rho=f(z,C(\rho(z,t))t)$, where $f$ is an arbitrary function and
$C(\rho)=v_{z}+c^{2}_{ls}(dv_{z}/d\rho)^{-1}/\rho$ is the local speed of
propagation. By a requirement that far from the perturbation the density is
equal to one and the velocity field is zero, the local speed is then given as
$C(\rho)=\pm c_{s}(6\rho^{1/5}-5)$, where the sign $\pm$ determines a
direction of propagation.

Figure 3:  (Color online) Space-time evolutions in the harmonic trap of the
atomic density for relatively weak repulsive potential strengthes
$|A|=V_{0}/\mu_{G}$ with the chemical potential $\mu_{G}=0.435\mu K$. The
atomic density integrated along the transverse direction is in units of
$10^{-2}/\mu m$ per particle. Panels (a) and (b) correspond to $|A|=0.0153$
and $0.23$, respectively. Left: the actual density. Right: the normalized
density (the actual density minus the ground-state density).

The repulsive potential produces an initial density
$\rho(z,0)=(1+A\exp({-z^{2}/\sigma^{2}}))^{5/2}$, where the magnitude of $A$
is proportional to the strength, i.e. $|A|\varpropto V_{0}$. After the
repulsive potential is suddenly turned off or switched on, the Gaussion shaped
perturbation breaks into two separate parts with half-amplitude of the initial
perturbation moving in the opposite directions sal2011 ; dam2004 ; dam12004 .
The extrema of the traveling wave has a constant amplitude $1+\eta$, and its
speed is given by

$c(\eta)=c_{s}[6(1+\eta)^{1/5}-5],$ (15)

where the wave amplitude is determined by the strength of the repulsive
potential through the relation $\eta=[(1+A)^{5/2}-1]/2$ dam12004 . It is seen
that the speed of shock waves explicitly depends on initial conditions. In the
limit $A\sim 0$, the speed reduces to the sound speed $c_{s}$. For $A<0$
($A>0$) corresponding to negative (positive) perturbations, one has subsonic
(supersonic) waves.

Figure 4:  (Color online) Space-time evolutions in the harmonic trap of the
atomic density for repulsive potential strengthes $|A|>1$. The atomic density
integrated along the transverse direction is in units of $10^{-2}/\mu m$ per
particle. Panels (a), (b) and (c) correspond to $|A|=1.22$, $6.11$, and
$14.6$, respectively. Panel (d) shows the experimental case $|A|=29.2$.

In the above analytical discussion, we assumed that the transverse density is
approximated by the TF ground-state, which means that the dynamics in the
transverse direction is frozen and no instability occurs as studied in Refs.
sal2011 ; low2013 ; dam2004 . Salasnich obtained the propagation speed of the
shock waves proportional to $[4(1+\eta)^{1/3}-3]$ in a different configuration
sal2011 , i.e. 3D homogenous unitary Fermi gas. Such expression differs from
our result only for the constants, because the integration of the transversal
harmonic confinement leads the power of the nonlinear term to change from 2/3
to 2/5. Therefore, one can predict the speed of the shock waves in a
homogeneous BEC in proportion to $[3(1+\eta)^{1/2}-2]$ dam2004 , due to the
nonlinear power $2/2$. Based on the effective 1D hydrodynamic equations (8)
incorporating the quantum pressure term (4), Lowman and Hoefer obtained
implicit relations for the speeds of supersonic shock waves low2013 . However,
numerical simulation in the following will show that the analytical result is
only valid for relatively weak repulsive potentials. As the strength of the
potential increases, the dispersive shock wave is created with larger
amplitude of oscillating wave-trains. In a strongly interacting Fermi gas,
they are easily subject to the transverse instability that leads to the
formation of vortex rings bul2014 ; cet2013 ; mun2014 ; wen2013 . Such decay
process eventually takes an significant effect on the dynamical properties of
shock waves.

###  III.2 numerical results

In the following we numerically solve the order-parameter equation to study
the formation and dynamics of shock waves in the harmonic trap, which are
initiated by different strengthes of the repulsive potential under the
experimental conditions. Due to the perturbations are created by suddenly
turning off the repulsive potential, they are negative and subsonic. The
amplitude of the perturbation is then expressed by $\eta=[(1-|A|)^{5/2}-1]/2$,
where the magnitude $|A|=V_{0}/\mu_{G}$ is characterized by the ratio of
strength of the repulsive potential to chemical potential, and
$\mu_{G}=0.435\;\mu K$ is obtained by substituting the experimental parameters
into Eq. (10).

Figure 3 shows the space-time evolutions of the atomic density integrated
along the transverse direction at relatively small repulsive potential
strengthes, and other parameters are the same as the experimental parameters.
Fig. 3(a) shows the case of a very small perturbation $|A|=0.0153$. However,
in the left panel the propagation of the wave cannot be directly observed
through the density profile. In order to visualize the trajectory of the wave,
we present the normalized density profile in the right panel, i.e. the actual
density subtracts the ground-state density. It is clear that the initially
induced negative perturbation splits into two density dips, which propagate
outward at an almost constant speed and slow down when approaching the
superfluid boundary due to the density dependence of the speed. For such weak
perturbation $|A|\sim 0$, one might expect to excite linear wave. One can
extract the propagation speed from the center of the normalized density
profile jos2007 ; sid2013 . The obtained speed of $0.347v_{F}$ is consistent
with the analytical prediction $c_{s}=\sqrt{\xi^{1/2}/5}v_{F}=0.351v_{F}$.
With increasing $|A|=0.23$ in Fig. 3(b), in the left panel we observe
directly the formation of the shock wave from the density profile, in which
the shock front is represented as a sharp discontinuity of the two density
lines forming a “V” shape. In addition, one can see that after a long time
dynamics, the steepness of the shock wave front is associated with oscillatory
behaviors, which are more obvious from the normalized density profile in the
right panel.

Figure 5: (Color online) Propagation speed of the shock wave fronts as a
function of the strength of the repulsive potential. Left insert: the
comparison of the numerical simulations ($\Box$) and the analytical prediction
Eq.(15) (dashed line). Right inset: the comparison of the results of the
experiment ($\diamondsuit$) jos2011 , the density functional theory
($\bigtriangleup$) bul2012 , and our results calculated for the cases in the
harmonic trap ($\Box$) and after an additional expansion (filled $\Box$).

In Figure 4, we show the space-time evolutions of the atomic density initiated
by large potential strengthes $|A|>1$, which means that the repulsive
potential results in two spatially separated clouds. Fig. 4(a) corresponds to
the case of $|A|=1.22$. Different from the cases of initial small
perturbations in Fig. 3, the density oscillation behaivors begin to occur in
the overlap of the two clouds at the center of the system. For larger values
of the strengthes in Fig. 4(b) ($|A|=6.11$) and Fig. 4(c) ($|A|=14.6$),
stronger oscillations of the densities in more extensive regions are observed.
But interestingly, from panels (a) to (d) we notice that the simulations
present very similar characteristic of the “V” shape. Fig. 4(d) shows the
experimental situation $|A|=29.2$, which is in agreement with the experimental
results [see Fig. 1 of Ref. bul2012 ]. Noticed that compared to the
experimental observation in Fig. 1 of Ref. bul2012 , which is on an expanded
scale realized by an additional $1.5\;{\rm ms}$ expansion after evolving in
the harmonic trap ( i.e. all three steps described in Sec. IIB are included),
in Fig. 3 and 4 we present the space-time evolutions in the harmonic trap
without the additional expansion (i.e. only the first two steps are included).

From Fig. 3 and 4 we not only observe the transition from a sound wave to
shock waves, but we also extract the propagation speeds of the shock wave
fronts by calculating the slopes of the two lines in the “V” shape. The
propagation speed of the shock waves as a function of the repulsive potential
strength is plotted in Figure 5. Two approximate regimes can be distinguished:
In the relatively small strength regime, the propagation speed decreases below
the sound speed as the strength increases. In the left inset we compare the
numerical results ($\Box$) to the analytical prediction Eq. (15) (dashed
line). One can find that the analytical expression agrees very well with the
numerical results in the very small strength regime ($|A|<0.2$). In this
regime, the transversal dynamics has a negligible effect, so that Eq. (15) is
an accurate expression for the speed of the shock waves derived from the
effective 1D model. Specifically, one can see that for $|A|>0.3$ the speed
presents a significant deviation from the analytical prediction, which
indicates that the 1D model is not adequate in the description of the 3D
system. The other interesting regime is $|A|>1$ corresponding to the cases of
Fig. 4, we find the propagation speed of the wavefront is independent on the
value of the strength, and remains about $0.24v_{F}$. The physical reason for
the unchanged speed can be explained as follows. For the regime of $|A|>1$,
the shock wave is formed by colliding two initial separated clouds. As these
two clouds are close each other and overlap gradually at the center of the
trap, the dispersive effect results in the formation of soliton trains. Due to
the transverse instability, the soliton trains quickly decay into vortex rings
and the density bulge peak is formed by filling with the numerous vortex
rings. We calculated the propagation speed of the shock waves from the
movements of self-steepen edges of the bulge peak. Therefore, the obtained
speeds actually correspond to the same expansion speed of the proliferation of
the vortex rings.

In the right inset, we show the numerical results for the experimental
situation $|A|=29.2$ calculated from two different situations. One denoted by
open square is obtained from the density profiles after evolving in the
harmonic trap (see Fig. 4(d)), while the other by filled square is calculated
from the density profiles after expansion for 1.5 ms after evolving in the
harmonic trap as performed in the experiment (see Fig. 1). The speed of
$0.302v_{F}$ obtained from the profiles after the additional expansion is much
larger than $0.243v_{F}$ calculated in the trap. This is because such
expansion method enhances the difference of the wavefront positions at two
sequence times. The result ($0.25v_{F}$, triangle) based on the DFT bul2012
and the experimental data ($0.33v_{F}$, diamond) jos2011  are also shown for
comparison.

##  IV Conclusion

In conclusion, based on the order-parameter equation we have presented a
detailed study for the experiment on dynamics of shock waves in the unitary
superfluid Fermi gas performed by the Thomas’s group, which shows good
agreement with the experiment. The shock wave is studied by colliding two
spatially separated clouds. As the two clouds gradually overlap at the center
of the trap, a soliton train forms due to the dispersive effect provided by
the quantum pressure term. Due to the transverse instability, the soliton
train decays very quickly into a large number of vortex rings. The boxlike-
shaped density peak observed in the experiment is then interpreted by the
proliferation of the vortex rings. In addition, we have studied the mechanism
of transition from a sound wave to shock waves by calculating the speeds of
the wavefronts and given an explanation why the speed of the shock wave
observed in the experiment is so close to the sound speed. For a very small
strength of the initial repulsive potential, sound wave is created with the
propagation speed of $0.347v_{F}$. As the strength increases in the relatively
weak regime, the nonlinear effect leads to the formations of shock waves, and
the speed decreases below the sound speed as a scaling behavior. When the
strength is moderate, the transversal dynamics takes an effect to suppress the
decrease of the speed. For the large strength regime where the shock wave is
formed by colliding two separated clouds, the propagation speed of the
wavefront is independent on the strength and remains about $0.24v_{F}$. We
understand this speed as the expansion speed of the proliferation of the
vortex rings. Finally, our numerical simulation demonstrates that the
expansion image method results in calculating the propagation speed from
$0.243v_{F}$ to $0.302v_{F}$, which is very close to $0.33v_{F}$ of the
experimental observation.

###### Acknowledgements.

We would like to thank John Thomas and James Joseph for suggesting to study
the propagation speed of the shock wave fronts, and useful comments on the
manuscript. We also thank Han Pu for the critical comments on the manuscript.
W.W. thanks the hospitality of KITPC where part of the writing is done, and
useful discussions with Hui Zhai. This work is supported by the NSFC under
Grant (No. 11105039 and No. 11274092), and Fundamental Research Funds for the
Central Universities of China (Program No. 2012B05714 and No. 2014B11414).

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11institutetext: University of Maryland at College Park
11email<EMAIL_ADDRESS>
# COVID-19’s Unequal Toll: An assessment of small business impact disparities
with respect to ethnorace in metropolitan areas in the US using mobility data

Saad Mohammad Abrar  Kazi Tasnim Zinat  Naman Awasthi  Vanessa Frias-Martinez

###### Abstract

Early in the pandemic, counties and states implemented a variety of non-
pharmacological interventions (NPIs) focused on mobility, such as national
lockdowns or work-from-home strategies, as it became clear that restricting
movement was essential to containing the epidemic. Due to these restrictions,
businesses were severely affected and in particular, small, urban restaurant
businesses. In addition to that, COVID-19 has also amplified many of the
socioeconomic disparities and systemic racial inequities that exist in our
society. The overarching objective of this study was to examine the changes in
small urban restaurant visitation patterns following the COVID-19 pandemic and
associated mobility restrictions, as well as to uncover potential disparities
across different racial/ethnic groups in order to understand inequities in the
impact and recovery. Specifically, the two key objectives were: 1) to analyze
the overall changes in restaurant visitation patterns in US metropolitan areas
during the pandemic compared to a pre-pandemic baseline, and 2) to investigate
differences in visitation pattern changes across Census Block Groups with
majority Asian, Black, Hispanic, White, and American Indian populations,
identifying any disproportionate effects. Using aggregated geolocated cell
phone data from SafeGraph, we document the overall changes in small urban
restaurant businesses’ visitation patterns with respect to racial composition
at a granularity of Census Block Groups. Our results show clear indications of
reduced visitation patterns after the pandemic, with slow recoveries. Via
visualizations and statistical analyses, we show that reductions in visitation
patterns were the highest for small urban restaurant businesses in majority
Asian neighborhoods.

###### Keywords:

COVID-19 business impact Mobility data Ethnoracial analysis

##  1 Introduction

The outbreak of the COVID-19 pandemic brought unprecedented challenges to
communities worldwide, prompting the implementation of various non-pharmaco-
logical interventions (NPIs) aimed at limiting mobility and curbing the spread
of the virus [1, 6]. These interventions, such as nationwide lockdowns and
remote work strategies, underscored the crucial role of restricting movement
in containing the epidemic [10, 9]. While these measures were effective in
mitigating the health crisis, they also had significant repercussions for
businesses, particularly small establishments that form the backbone of local
economies [4, 3].

The pandemic has also exposed and magnified pre-existing socioeconomic
disparities and systemic inequities within our society. In fact, vulnerable
populations, including racial and ethnic minorities, have borne a
disproportionate burden of the pandemic’s impact including anti-Asian racism
[5] as well the condition of being frontline workers [8]. Understanding the
impact of COVID-19 on small businesses and its intersection with racial
composition is also vital for policymakers, community leaders, and
stakeholders. By identifying disparities and informing targeted interventions,
tangible work can be done towards fostering a more equitable recovery and
supporting the long-term sustainability of small businesses.

To better understand the implications of COVID-19 on small businesses and to
shed light on the intersection of ethnorace and business resilience, this
study leverages aggregated geolocated cell phone data from SafeGraph,
characterizing visits to small urban businesses in the US, with demographic
data related to the social composition of the areas where these businesses are
located. Analyzing business visitation patterns at the level of Census Block
Groups in the US, we aim to provide insights into the changes in small urban
business activity and their association with the race and ethnicity of their
tracts. Although many types of businesses were affected by the pandemic, we
focus on non-chain urban restaurants because these small businesses are major
contributors to job creation while also being more vulnerable to falling
demand [3].

The objective of this research is twofold. Firstly, we examine the overall
changes of restaurant visitation patterns following the onset of the pandemic,
assessing the extent to which small urban restaurant businesses in the US have
been affected and are recovering from the initial shock. Secondly, we
investigate whether disparities in visitation patterns exist across different
racial and ethnic groups, seeking to uncover potential inequities in the
recovery process across metropolitan areas. By delving into the visitation
patterns and changes in visitation volumes across restaurants, we can gain
valuable insights into the specific challenges faced by small urban restaurant
businesses in the US during the COVID-19 pandemic. This information will
contribute to a more nuanced understanding of the restaurant industry’s
dynamics and inform targeted strategies and support mechanisms for these
vulnerable businesses.

The rest of the paper is organized as follows. In Section 2, we present a
comprehensive description of the mobility and census data we will use in the
analysis. Section 3 describes the statistical and visualization methods we use
to explore changes in restaurant visitation patterns during the pandemic, and
to examine the relationship between ethnorace and visitation changes as well
as recovery patterns. Finally, Section 4 will present our main findings,
followed by Section 5 with a description of the proposed future work.

##  2 Data

###  2.1 Mobility Data

The mobility patterns of millions of anonymized users in the US were tracked
using mobile app software development kits (SDKs) by SafeGraph, a data
provider that made this information publicly available at the beginning of the
pandemic. To analyze the impact of COVID-19 on business visitation patterns,
we utilized the points of interest (POIs) dataset from SafeGraph. We focused
on two components of the POI data:

  * •

Core POI:  It includes the name and brand (indicating whether the POI is a
part of a chain)

  * •

Patterns:  It includes the volume of daily visits per identified POI in the
Safegraph database.

To focus specifically on small business restaurants, our analysis involves
extracting points of interest (POIs) that meet two criteria: they are not
associated with any chain (brand), and they fall within the category of
“Restaurants and Other Places”. Around 80% of POIs in SafeGraph have no brand
associated as they are single commercial locations. The visit patterns were
measured as changes in volumes compared to the traffic before the pandemic. We
collected SafeGraph data from January 1st, 2020 to January 31st, 2021. The
data collected between January 1st, 2020, and March 7, 2020, was defined as
the baseline to assess changes in visitation patterns during the pandemic,
which we mark with a starting date of March 15, 2020. To reveal long-term
trends that may be concealed by occasional fluctuations we utilized a 7-day
rolling average to smooth the time series data. This approach enables us to
identify significant changes and discern overall changes in the data,
providing a clearer picture of the long-term effects of the pandemic on small
business restaurant visitation patterns.

###  2.2 ACS Data

To evaluate disparities in the impact of COVID-19 on small urban restaurant
businesses with respect to race and ethnicity, we utilized three datasets
obtained from the American Census Survey datahub111https://data.census.gov/.
These datasets provided valuable information for our research:

  * •

CBG ShapeFiles: We utilized CBG ShapeFiles to gather the boundaries of the
Census Block Groups. We use CBG as the geographical unit of analysis,
assigning to each CBG the changes in volumes of visits to small restaurants
located within the CBG.

  * •

Urban-rural classification: This dataset helped us identify urban CBGs by
categorizing regions based on their urban level. We specifically focused on
CBGs classified as level 1, which corresponds to large metropolitan counties.

  * •

Race Distribution: This dataset provided detailed information on the racial
composition of the population at the Census Block Group (CBG) level. It
allowed us to examine the distribution of different racial groups within
specific geographic areas. To carry out the proposed analysis, each CGB was
associated to the majority group race i.e., the race with the largest
population volume in that CBG.

Figure 1 shows all the urban Census Block Groups representing large
metropolitan areas, providing an overview of the locations used in our
analysis. Figure 2 represents the count of CBGs associated to each majority
race. To delve deeper into the analysis, Figure 3 shows an example of the
racial composition across CBGs for the Bay area and for New York City.

Figure 1: Urban Census Block Groups representing all the large metropolitan
areas used in our analysis. Figure 2: Count of CBGs associated to each
majority race.

(a) Racial composition of CBGs in the Bay Area, Northern California.

(b) Racial composition of CBGs in and around New York City.

Figure 3: Racial composition at selected urban locations.

###  2.3 Why CBGs as a proxy measure of localized impact?

Firstly, CBG allows for a localized impact assessment. It provides a finer
level of granularity than larger geographic units, such as census tracts or
counties. By focusing on CBGs, researchers can zoom in on specific
neighborhoods or areas within a city or region. This localized approach
enables a more precise understanding of the effects of interventions,
policies, or demographic changes on smaller-scale populations.

Secondly, CBG is particularly useful when studying highly homogenous
populations. In some cases, neighborhoods or communities may exhibit a high
degree of homogeneity in terms of race, ethnicity, income level, or other
demographic characteristics. By analyzing CBGs within these areas, researchers
can gain insights into the dynamics and changes occurring within these
homogeneous populations. This level of analysis can help identify trends,
patterns, or disparities that may not be evident when examining larger, more
heterogeneous geographic units.

Lastly, the reliance on CBGs as the unit of analysis is prevalent in most
quantitative studies of neighborhood racial change [2, 11, 12]. Researchers
often utilize CBGs because they offer a balance between geographic precision
and sample size. CBGs generally have a population size that is manageable for
analysis while still providing enough data points to draw statistically
significant conclusions. This widespread usage of CBGs as a unit of analysis
allows for comparability and consistency across studies, facilitating the
accumulation of knowledge and the identification of broader trends.

##  3 Methodological Approach

For each urban CBG in the US, we compute a time series with the daily
percentage change in aggregated visits to small restaurants within that CBG.
The daily percentage change reflects changes in volumes of visits during the
pandemic with respect to the predefined pre-pandemic visitation patterns. To
investigate the recovery patterns across ethnoracial groups, we carry out
separate analyses for each race and ethnicity by exclusively looking into the
daily percentage change time series of census block groups with the same
ethnoracial majority. For instance, to examine the pattern specific to the
Asian demographic subgroup, we focus solely on the census clock groups - and
their corresponding daily visitation time series - where Asians constitute the
majority population.

The proposed methodological approach will allow us to examine overall changes
in restaurant visitation patterns following the pandemic, and whether
disparities in visitation patterns exist across racial and ethnic groups,
seeking to uncover potential inequities in the recovery process across
metropolitan areas. Nevertheless, a potential limitation of this approach is
that it will not reveal the reasons behind changes in the visitation patterns
i.e., small urban restaurants in specific ethnoracial CBGs could have received
fewer visits during the pandemic either because residents of that CBG visited
less their local restaurants, or because non-residents were visiting less that
CBG. While the former would point to the impact of a specific ethnorace’s
visiting behaviors to their local restaurants; the latter would signal changes
in visiting behaviors from non-residents to specific ethnoracial CBGs. In
addition, changes in visitation patterns for specific ethnoraces could also be
due to higher volumes of business closings.

To evaluate overall small urban restaurants’ recovery patterns, and changes
across ethnoracial groups, we will also analyze the average percentage change
in volumes of visits during the pandemic with respect to their corresponding
pre-pandemic baselines, and we carry out this analysis for each ethnoracial
group across three phases: phase I or social distancing (from mid March till
mid May), phase II or management of the pandemic post-onset (from mid May till
the end of November), and phase III or vaccination phase, to characterize the
period when vaccines became available for certain social groups (from November
onwards).

To evaluate statistical differences in recovery patterns across ethnoracial
groups, we compute a two-sample Kolmogorov-Smirnov test [7] between each pair
of ethnoracial average percentage change time series, with each average time
series computed as the average of all time series across CBGs of the same
majority ethnorace. The KS test, which is non-parametric, is used to capture
statistically significant similarities and differences in both location and
scale between pairs of ethnoracial time series. This focused approach allows
us to analyze and compare the recovery trajectories among various racial
communities. By employing this method, our objective is to identify any
disparities or variations in post-pandemic recovery patterns based on racial
composition within urban settings.

##  4 Results

Figure 4 shows heatmap matrices computed with the percentage change time
series for the CBGs in our analysis. Each CBG is represented by a time series
from January 7, 2020 to January 31, 2021; and every CBG contributes to a
single pixel per day in the heatmap matrix. The heatmap matrix’s density
reflects the combined contribution of each CBG, enabling a comprehensive
visual representation of the variations in visitation patterns over time and
across ethnoraces. While 4(a) displays the heatmap matrix for all the CBGs,
Figures 4(b–f) show the percentage change time series represented as heatmap
matrices for each ethnicity or race in our analysis. Figure 5, on the other
hand, represents the average percentage change time series across all CBGs
with the same majority race. In other words, for each heatmap matrix in
Figures 4(b–f), we compute its average time series, which allows for a
straight forward comparison of changes in restaurant visitation patterns
across ethnoraces.

(a) Heatmap matrix for CBGs across all groups

(b) Heatmap matrix for CBGs with American Indian Majority

(c) Heatmap matrix for CBGs with Asian Majority

(d) Heatmap matrix for CBGs
with Black Majority

(e) Heatmap matrix for CBGs with Hispanic Majority

(f) Heatmap matrix for CBGs with White Majority

Figure 4: Heatmap matrices for percentage changes in visit patterns

As we can observe in Figure 4, right after the global onset of the pandemic
(March 15, 2020) the overall volume of visits to small urban restaurants as
well as the volumes of visits disaggregated by ethnorace heavily decreased and
did not go back to normal even after one year into the pandemic. To analyze
disparities in visitation patterns and trajectory recoveries across
ethnoraces, Figure 5 shows one time series per ethnorace, representing the
average percentage changes in visitation patterns for small restaurants
located in CBGs of a specific ethnorace. Overall, we observe that all CBGs,
independently of their majority race were heavily affected by the pandemic
with large percentage decreases of visitation patterns when compared to the
baseline. We also observe that all time series reflect smooth trends across
phases except for the American Indian time series, that suffers from more
oscillations probably due to the smaller number of majority CBGs of that race
in our dataset. Despite the large pandemic impact on visitation patterns,
there exist nuanced differences across ethnoraces, with majority Asian and
majority American Indian being not only the most impacted, but also the ones
with the slowest recovery even after one year into the pandemic.

Figure 5: Average percentage change time series across all CBGs for each
ethnicity or race. The vertical dashed lines separate the pre-pandemic phase
from phases 1, 2 and 3.

To quantify these differences, Table 1 displays the average percentage changes
in visitation patterns per ethnorace and across the three phases explained
earlier. We can observe that Asian and American Indian CBGs are the ones most
impacted, with larger average percentage decreases in visits with respect to
pre-pandemic times of $42.7\%$ and $41.7\%$, respectively, across all phases;
reaching maximum decreases of $63\%$ and $62\%$ respectively for phase 1
(onset of the pandemic). Table 2 shows the KS statistics and p-values for all
combinations of ethnoracial average percentage change time series. The null
hypothesis was rejected for all pairs of distinct time series ($p\\_value=0$)
and the Asian average percentage change time series is the one most different
to the others (with the highest ks statistic values). Put together with the
findings in Table 1, this statistical analysis shows that Asian CBGs are not
only the least visited across pandemic phases, but that they are also the most
statistically significantly different from the other ethnorace’s
distributions, with ks statistics in the $0.48-0.58$ range when compared to
majority Hispanic, White and Black CBGs, albeit more similar to the American
Indian time series ($ks=0.22$).

| American Indian | Asian | Black | Hispanic | White
---|---|---|---|---|---
Phase 1 | -61.9199 (11.42972) | -63.0853 (11.52541) | -49.3982 (9.574766) | -54.6981 (10.45263) | -60.1125 (11.40951)
Phase 2 | -44.7563 (8.186457) | -47.9976 (4.229237) | -36.8484 (4.287543) | -41.2918 (3.107061) | -37.1945 (5.706389)
Phase 3 | -57.1392 (4.839321) | -51.597 (1.347966) | -43.0362 (2.738588) | -44.043 (2.244181) | -43.8665 (3.330658)
All Phases | -41.7721 (21.8725) | -42.709 (20.79114) | -33.4509 (16.75843) | -36.7284 (18.04656) | -35.4661 (19.03601)

Table 1: Average percentage change and the standard deviation (std) in
visitation patterns per ethnorace over all the time period under study as well
as for each phase.

KS_Test | American Indian | Asian | Black | Hispanic | White
---|---|---|---|---|---
American Indian | (0.0, 1.0) | (0.21995, 0.0) | (0.38107, 0.0) | (0.30179, 0.0) | (0.31969, 0.0)
Asian | (0.21995, 0.0) | (0.0, 1.0) | (0.58312, 0.0) | (0.48593, 0.0) | (0.50639, 0.0)
Black | (0.38107, 0.0) | (0.58312, 0.0) | (0.0, 1.0) | (0.27877, 0.0) | (0.10997, 0.01761)
Hispanic | (0.30179, 0.0) | (0.48593, 0.0) | (0.27877, 0.0) | (0.0, 1.0) | (0.32481, 0.0)
White | (0.31969, 0.0) | (0.50639, 0.0) | (0.10997, 0.01761) | (0.32481, 0.0) | (0.0, 1.0)

Table 2: Two-sample Kolmogorov-Smirnov tests between each pair of ethnoracial
average percentage change time series. For each pair of distributions we
report the ks statistic and its significance (p-value).

##  5 Future Directions

The initial results point to interesting insights with respect to impact and
recovery visitation patterns across ethnoraces. Nevertheless, further analysis
is necessary to obtain more conclusive results. Several approaches can be
undertaken to enhance the analysis and derive meaningful insights. These
include:

  * •

Disaggregating CBGs by County: Instead of analyzing all the CBGs together, and
aggregate by ethnorace, conducting an analysis that focuses on individual
counties can provide more localized and targeted insights. This approach
allows for a deeper understanding of the variations and patterns within
specific geographic regions. In addition to that, examining and reporting
statistically disproportionate behaviors per county can shed light on any
significant deviations from the expected patterns. Furthermore, analysis can
help identify counties where the impact on small urban restaurant businesses’
visitation patterns significantly differ from others, indicating potential
underlying factors influencing recovery.

  * •

Analysis of Policy Effects: Analyzing the impact of different policies enacted
during the pandemic on small urban restaurant visitation patterns can provide
insights into how specific regulations influenced small businesses. This
analysis can involve assessing the effects of measures like lockdowns,
capacity restrictions, or curfews on the visitation trends and recovery
trajectories.

  * •

Regression Analysis: Conducting regression analysis on the percentage change
in visits, with specific demographic variables as independent features, can
help determine the relationships and influences between demographic factors
and visitation patterns. Therefore, this analysis can provide a more nuanced
understanding of how factors like racial composition, income levels, or
education levels might have an effect on small urban business visitation
patterns during and after the pandemic.

  * •

Causal Analysis of Changes in Visitation Patterns: Besides the analysis of the
statistical changes in visitation patterns presented, it would also be
important to understand the causes behind the changes. Visitation patterns can
change because internal CBG behaviors change, pointing to specific ethnoracial
communities responding differently to the pandemic; because external CBG
behaviors change, pointing to how other ethnoraces might change their visits
to small restaurants located in specific ethnoracial CBGs; or because small
restaurants located in CBGs of a specific ethnorace might be closing at higher
rates. Identifying the role of all of these factors is critical to inform
targeted strategies and support mechanisms for vulnerable businesses.

By incorporating these approaches into the analysis, we posit we can obtain
more conclusive and robust findings. These methods allow for a comprehensive
examination of the data and will enhance our understanding of the factors
influencing small urban restaurants’ visitation patterns during the pandemic.

## References

  * [1] Baker, R.E., Park, S.W., Yang, W., Vecchi, G.A., Metcalf, C.J.E., Grenfell, B.T.: The impact of covid-19 nonpharmaceutical interventions on the future dynamics of endemic infections. Proceedings of the National Academy of Sciences 117(48), 30547–30553 (2020)
  * [2] Brough, R., Freedman, M., Phillips, D.C.: Understanding socioeconomic disparities in travel behavior during the covid-19 pandemic. Journal of Regional Science 61(4), 753–774 (2021)
  * [3] Cowling, M., Liu, W., Ledger, A., Zhang, N.: What really happens to small and medium-sized enterprises in a global economic recession? uk evidence on sales and job dynamics. International Small Business Journal 33(5), 488–513 (2015)
  * [4] Elimam, H., et al.: The role of small businesses (small scale economic projects) in alleviating the acuity of unemployment. International Business Research 10(3), 120–132 (2017)
  * [5] Huang, J.T., Krupenkin, M., Rothschild, D., Lee Cunningham, J.: The cost of anti-asian racism during the covid-19 pandemic. Nature Human Behaviour pp. 1–14 (2023)
  * [6] Kuo, S.C., Shih, S.M., Chien, L.H., Hsiung, C.A.: Collateral benefit of covid-19 control measures on influenza activity, taiwan. Emerging infectious diseases 26(8), 1928 (2020)
  * [7] Marozzi, M.: Nonparametric simultaneous tests for location and scale testing: a comparison of several methods. Communications in Statistics-Simulation and Computation 42(6), 1298–1317 (2013)
  * [8] OECD: The unequal impact of covid-19: a spotlight on frontline workers, migrants and racial/ethnic minorities (2022)
  * [9] Oh, J., Lee, H.Y., Khuong, Q.L., Markuns, J.F., Bullen, C., Barrios, O.E.A., Hwang, S.s., Suh, Y.S., McCool, J., Kachur, S.P., et al.: Mobility restrictions were associated with reductions in covid-19 incidence early in the pandemic: evidence from a real-time evaluation in 34 countries. Scientific reports 11(1), 1–17 (2021)
  * [10] Oka, T., Wei, W., Zhu, D.: The effect of human mobility restrictions on the covid-19 transmission network in china. PloS one 16(7), e0254403 (2021)
  * [11] Ong, P., Comandom, A., DiRago, N., Harper, L.: Covid-19 impacts on minority businesses and systemic inequality (2020)
  * [12] Reibel, M., Regelson, M.: Neighborhood racial and ethnic change: The time dimension in segregation. Urban Geography 32(3), 360–382 (2011)


# On the set of Kronecker numbers

Sayan Goswami  The Institute of Mathematical Sciences, A CI of Homi Bhabha
National Institute, CIT Campus, Taramani, Chennai 600113, India.
<EMAIL_ADDRESS>,  Wen Huang  School of Mathematical Sciences, University
of Science and Technology of China, Hefei 230026, P. R. China.
<EMAIL_ADDRESS>and  XiaoSheng Wu  School of Mathematics, Hefei
University of Technology, Hefei 230009, P. R. China<EMAIL_ADDRESS>
###### Abstract.

An positive even number is said to be a Kronecker number if it can be written
in infinitely many ways as the difference between two primes, and it is
believed that all even numbers are Kronecker numbers. We will study the
division and multiplication of Kronecker numbers to study the largeness of the
set of Kronecker numbers. A numerical lower bound for the density Kronecker
numbers among even numbers is given, and it is proved that there exists a
computable constant $k$ and a set $D$ consisting of at most 720 computable
Maillet numbers such that, for any integer $n$, $kn$ can be expressed as a
product of a Kronecker number with a Maillet number in $D$. Meanwhile, it is
proved that every positive rational number can be written as a ratio of two
Kronecker numbers.

###### Key words and phrases:

difference of primes; Kronecker number; $\Delta_{r}^{*}$-set; IP-set.

###### 2010 Mathematics Subject Classification:

11N05, 37A44

##  1\. Introduction

The distribution of the difference of primes is a widely concerned theme in
number theory, and the following conjecture is well-known.

######  Conjecture 1.1 (Kronecker [11]).

Every even number can be written in infinitely many ways as the difference of
two primes.

The conjecture is beyond the reach of modern mathematics. However, series of
recent breakthroughs on the twin prime conjecture indicate that there is an
integer not exceeding than 246, which can be written in infinitely many ways
as the difference of two primes; see also [2], [15], [12], and [14].

######  Definition (Maillet number, Kronecker number).

An even number $n$ is called a Maillet number (Kronecker number), if it can be
written (in infinitely many ways) as the difference of two primes.

We apply $\mathcal{K}$ for the set of all Kronecker numbers. The twin prime
conjecture is about the lower bound of $\mathcal{K}$. Another important aspect
of the Kronecker conjecture is how “large” the set $\mathcal{K}$ is. It is
proved by Pintz [13] that $\mathcal{K}$ is a syndetic set, which was also
obtained by Granville, Kane, Koukoulopoulos and Lemke Oliver [4], using a
different method later.

One kind of sets larger than the syndetic set are known as the
$\Delta_{r}^{*}$-set.

######  Definition ($\Delta_{r}^{*}$-set).

Let $S$ be a non-empty subset of $\mathbb{N}$, and we define its difference
set $\Delta(S)$ via

$\Delta(S)=(S-S)\cap\mathbb{N}=\\{a-b:a,b\in S,a>b\\}.$

A subset $A$ of $\mathbb{N}$ is called a $\Delta_{r}^{*}$-set if for every
subset $S$ of $\mathbb{N}$ with $|S|\geq r$, there is

$A\cap\Delta(S)\neq\emptyset.$

In [10], Huang and Wu have proved that $\mathcal{K}$ should be a
$\Delta_{r}^{*}$-set.

######  Theorem 1.2.

The set $\mathcal{K}$ of Kronecker numbers is a $\Delta_{r}^{*}$-set for any
$r\geq 721$.

It is also mentioned in [10] that the number $721$ can be sharpen to $19$ if
the primes have level of distribution $\theta$ for every $\theta<1$. Here, for
some given $\theta>0$, we say the primes have ‘level of distribution $\theta$’
if, for any $W>0$, it holds that

$\sum_{q\leq
x^{\theta}}\max_{(a,q)=1}\Big{|}\pi(x;q,a)-\frac{\pi(x)}{\phi(q)}\Big{|}\ll_{W}\frac{x}{(\log
x)^{W}}.$

In this work, we try to obtain more information on how large $\mathcal{K}$ is,
and our deduction starts from Theorem 1.2. Our first result is a numerical
lower bound for the density of $\mathcal{K}$ among even numbers.

######  Theorem 1.3.

Let $\alpha$ be the density of Kronecker numbers among all positive even
numbers. We have $\alpha\geq\frac{1}{360}\prod_{p\leq 720}(1-p^{-1})$. If the
primes have level of distribution $\theta$ for every $\theta<1$, we have
$\alpha\geq\frac{1024}{51051}$.

Our next result is about the representation of integers by products of
differences among primes, which is motivated by the following question, asked
by Fish in [1].

######  Question.

For a given infinite set $E\subset\mathbb{Z}$, how much structure does the set
$(E-E)\cdot(E-E)$ possess ?

Fish [1] considered the question with $E$ being positive density subset of
$\mathbb{Z}$. Using Furstenberg’s correspondence principle, he proved that
there exist $k_{0}$ (depending on the densities of $E_{1}$ and $E_{2}$) and
$k\leq k_{0}$ that

$k\mathbb{Z}\subset(E_{1}-E_{1})\cdot(E_{2}-E_{2}).$

It is naturally to consider the question with $E$ being the set of primes, and
this case is not contained by Fish’s work since the set of primes is an
infinite set of $\mathbb{Z}$ but not owning a positive upper Banach density.
Based on Theorem 1.2, Goswami[3] extends Fish’s result to the case about
primes that

(1.1) $\displaystyle
k\mathbb{Z}\subset(\mathbb{P}-\mathbb{P})\cdot(\mathbb{P}-\mathbb{P}).$

We can say something more for the case about primes. In detail, on the right-
hand side of (1.1), one factor can be restricted to a finite subset of
$\mathbb{P}-\mathbb{P}$, consisting of 720 Maillet numbers, and the other
factor takes value among Kronecker numbers. We present our main result in the
following.

######  Theorem 1.4.

There exist a computable constant $k$ and a set $D$, consisting of at most 720
computable Maillet numbers, that

$k\mathbb{Z}\subset D\cdot\mathcal{K}.$

The proof of Theorem 1.4 would be based on Theorem 1.2 as well as a recent
work on linear equations in primes by Green, Tao, and Ziegler. The number
$720$ here would also be sharpen to $18$ if the primes have level of
distribution $\theta$ for every $\theta<1$.

To see how large $\mathcal{K}$ is, we also consider the ratio of two Kronecker
numbers. The following theorem is about the division of the set $\mathcal{K}.$

######  Theorem 1.5.

Every positive rational number can be written as a ratio of two elements from
$\mathcal{K}.$

##  2\. The density of Kronecker numbers

In this section, we will deduce a lower bound for the density of a general
$\Delta_{r}^{*}$-set. As a corollary, we will prove Theorem 1.2.

######  Lemma 2.1.

Let $H$ be a $\Delta_{r}^{*}$-set, and denote by

$A_{r}(a)=\\{a,2a,\dots,(r-1)a\\}.$

Then, for any integer $a>0$,

$H\cap A_{r}(a)\neq\emptyset.$

###### Proof.

Just taking $S=\\{a,2a,\dots,ra\\}$, one has

$\Delta(S)=\\{a,2a,\dots,(r-1)a\\}.$

Thus, the lemma follows immediately from the definition of
$\Delta_{r}^{*}$-set.

######  Theorem 2.2.

If $H$ is a $\Delta_{r}^{*}$-set, we have

$\frac{H\cap[1,N]}{N}\geq\prod_{p\leq r-1}\left(1-p^{-1}\right)+o(1).$

###### Proof.

By Lemma 2.1, every $A_{r}(a)$ contains at least one element of $H$. We may
obtain a lower bound for the cardinality of $H$ by counting the number of the
sets $A_{r}(a)$, which are disjoint with each other. If $a<b$ be two integers
with $A_{r}(a)\cap A_{r}(b)\neq\emptyset$, then there are two integers $i,j$,
with $(i,j)=1$ and $1\leq i<j\leq r-1$, satisfying

$\frac{a}{b}=\frac{i}{j}.$

Thus, $A_{r}(a)$ should disjoint with each other if we only counter over such
$a$, which does not contain any prime less than $r-1$. For sufficiently larger
$N$, it is obvious that

$\left|\left\\{a:\Bigg{(}a,\prod_{p\leq
r-1}p\Bigg{)}=1,A_{r}(a)\subset[1,N]\right\\}\right|=\frac{N}{r}\prod_{p\leq
r-1}(1-p^{-1})+O(1).$

Now the theorem follows immediately.

###### Proof of Theorem 1.3.

Theorem 1.3 follows immediately from Theorems 1.2 and 2.2 by taking $r=721$
and $r=19$ respectively.

##  3\. Representation of integers

In this section, we give the proof of Theorem 1.4. In the deduction of the
constant $k$ as well as the set $D$, we appeal to the recent work on linear
equations in primes by Green, Tao, and Ziegler.

###  3.1. Linear equations in primes

We present some necessary detail in this section, and more contents about the
materials could be found in [5]. Let $d,t$ be integers. A system of affine-
linear forms on $\mathbb{Z}^{d}$ is a collection
$\Psi=\\{\psi_{1},\dots,\psi_{t}\\}$ with
$\psi_{i}:\mathbb{Z}^{d}\rightarrow\mathbb{Z}$ being affine-linear forms. If
$N>0$, the _size_ $\parallel\Psi\parallel_{N}$ of $\Psi$ relative to the scale
$N$ is the quantity

$\parallel\Psi\parallel_{N}:=\sum_{i=1}^{t}\sum_{j=1}^{d}|\dot{\psi}_{i}(e_{j})|+\sum_{i=1}^{t}\left|\frac{\psi_{i}(0)}{N}\right|,$

where

$\dot{\psi}_{i}(e_{j})=\psi_{i}(e_{j})-\psi_{i}(0)$

with $e_{1},e_{2},\dots,e_{d}$ being the standard basis for $\mathbb{Z}^{d}$.
For a system $\Psi$, its _local factor_ $\beta_{p}$ for a prime $p$ is defined
via

(3.1)
$\displaystyle\beta_{p}:=\frac{1}{p^{d}}\sum_{n\in\mathbb{Z}_{p}^{d}}\prod_{i=1}^{t}\Lambda_{\mathbb{Z}_{p}}(\psi_{i}(n)),$

where $\mathbb{Z}_{p}=\\{0,1,\dots,p-1\\}$ is the residue class of integers
modulo $p$, and $\Lambda_{\mathbb{Z}_{p}}(n)$ is the _local von Mangoldt
function_ defined by

$\Lambda_{\mathbb{Z}_{p}}(n)=\begin{cases}\frac{p}{p-1},&if\ (n,p)=1;\\\
0,&otherwise.\end{cases}$

######  Definition (Complexity).

The complexity of the $\Psi$ is the least integer $s$ that, for each
$\psi_{i}$, one can cover the $t-1$ forms $\\{\psi_{j}:j\neq i\\}$ by $s+1$
classes, such that $\psi_{i}$ does not lie in the affine-linear span of any of
these classes; if no such $s$ exists, we say that the complexity is $\infty$.

Main results of [6] and [7] indicate the following theorem.

######  Theorem 3.1 (Green, Tao, and Ziegler).

Let $N,d,t,L$ be positive integers, and let
$\Psi=\\{\psi_{1},\dots,\psi_{t}\\}$ be a system of affine-linear forms with
size $\parallel\Psi\parallel_{N}\leq L$. Let $K\subset[-N,N]^{d}$ be a convex
body. If $\Psi$ is finite complexity, we have

$\\#\left\\{n\in K\cap\mathbb{Z}^{d}:\psi_{1}(n),\dots,\psi_{t}(n)\ \
\text{prime}\right\\}=\left(1+o_{t,d,L}(1)\right)\frac{\beta_{\infty}}{\log^{t}N}\prod_{p}\beta_{p}+o_{t,d,L}\left(\frac{N^{d}}{\log^{t}N}\right),$

where $\beta_{\infty}:=vol_{d}\left(K\cap\Psi^{-1}(\mathbb{R}^{+})^{t}\right)$
is typically of size $N^{d}$, and where the singular product
$\prod_{p}\beta_{p}$ is always convergent.

It has been pointed out in [5]that the singular product $\prod_{p}\beta_{p}$
is always convergent, but it may still vanish since $\beta_{p}=0$ is possible
for small $p$. So, we should note that the conjecture works only for the
system of affine-linear forms with $\beta_{p}\neq 0$, for all $p$
(essentially, for small $p=O_{t,d,L}(1)$ is enough).

For the case with complexity $s>2$, the theorem was first proved under an
assumption that the inverse Gowers-norm conjecture and the Möbius and
nilsequences conjecture are true. And then these two conjectures were proved
to be true by them in combination with Ziegler; see also [6] and [7].

###  3.2. Proof of Theorem 1.4

To prove our theorem, we will appeal to a special case of Theorem 3.1, which
we provide in the following lemma.

######  Lemma 3.2.

Let $\Psi=\\{\psi_{1},\dots,\psi_{t}\\}$ be a system of affine-linear forms of
finite complexity with $\psi_{i}:\mathbb{Z^{+}}^{d}\rightarrow\mathbb{Z^{+}}$,
$1\leq i\leq t$, and $\beta_{p}\neq 0$ for any prime $p$. There are infinitely
many lattice points $n\in{\mathbb{Z}^{+}}^{d}$, which make all $\psi_{i}(n)$
prime.

###### Proof.

Note in Theorem 3.1 that we can take $K=[-N,N]^{d}$ here. For
$\psi_{i}:\mathbb{Z^{+}}^{d}\rightarrow\mathbb{Z^{+}}$, we have

$\beta_{\infty}:=vol_{d}\left(K\cap\Psi^{-1}((\mathbb{R}^{+})^{t}\right)\geq
vol_{d}\left([-N,N]^{d}\cap{\mathbb{Z}^{+}}^{d}\right)\geq N^{d}.$

Also, the singular product $\prod_{p}\beta_{p}$ would not vanish here since
$\beta_{p}\neq 0$ for all $p$. Thus, the asymptotic formula in Theorem 3.1
owns a dominated main term, and the lemma follows immediately.

For each $j=1,2,\dots,720$, we apply the coefficients $a_{j}$, which are
defined via

$a_{j}=\frac{720!}{j}.$

We consider a system of affine-linear forms
$\Psi=\\{\psi_{1},\dots,\psi_{1440}\\}$ via

$\displaystyle\psi_{2j-1}(n_{1},\dots,n_{720},m)=n_{j},$
$\displaystyle\psi_{2j}(n_{1},\dots,n_{720},m)=n_{j}+a_{j}m$

for $j=1,\dots,720$. This is a special system of affine-linear forms with
complexity $s=1$, and it is obvious that
$\psi_{i}:\mathbb{Z^{+}}^{d}\rightarrow\mathbb{Z^{+}}$ since all coefficients
are positive integers.

To apply Lemma 3.2, we also need $\beta_{p}\neq 0$. By (3.1) this would be
available if, for each $p$, one can find a lattice point
$n\in\mathbb{Z}_{p}^{d}$ that $(\psi_{i}(n),p)=1$ holds for all $i$.
Obviously, the lattice point $n=(1,1,\cdots,1,0)\in\mathbb{Z}_{p}^{721}$ is
eligible for our system. Thus, by Lemma 3.2, there are infinitely many lattice
points $(n_{1},\dots,n_{720},m)\in{\mathbb{Z}^{+}}^{721}$, which make all
$\psi_{i}$ prime. That is to say, for each $m$ in these lattice points, the
set $\\{a_{1}m,a_{2}m,\dots,a_{720}m\\}$ is consisting of Maillet numbers. We
apply $m^{\prime}$ for the least one of these $m$, which is a computable
number since the system of affine-linear forms is specific.

To prove the theorem, we take the constant

$k=720!m^{\prime},$

and let the set be

$D=\\{a_{1}m^{\prime},a_{2}m^{\prime},\dots,a_{720}m^{\prime}\\}.$

Also, for any integer $b>0$, Theorem 1.2 indicates that there is at least a
Kronecker number in the set $\\{b,2b,\dots,720b\\}$. If $jb$ with $1\leq j\leq
720$ is a Kronecker number, we can write

$kb=a_{j}m^{\prime}\cdot jb\in D\cdot\mathcal{K},$

which establishes Theorem 1.4.

##  4\. Representation of rationals

In this section we will use arguments from Ramsey theory to prove Theorem 1.5.
First we will prove three lemmas and then a more general Theorem 4.5. As a
corollary we will prove our Theorem 1.5.

The notion of $IP$ sets and $IP_{r}$ sets are well studied in Ramsey theory.
Let $\mathcal{P}_{f}\left(\mathbb{N}\right)$ be the collection of nonempty
finite subsets of $\mathbb{N}.$

######  Definition.

A set $A\subset\mathbb{N}$ is said to be an $IP$ (resp. $IP_{r}$ for some
$r\in\mathbb{N}$) set if there exists a sequence $\langle
x_{n}\rangle_{n\in\mathbb{N}}$ (resp. $\langle x_{n}\rangle_{n=1}^{r}$ ) such
that $A=FS\left(\langle x_{n}\rangle_{n\in\mathbb{N}}\right)$ (resp.
$A=FS\left(\langle x_{n}\rangle_{n=1}^{r}\right)$).

A set is said to be an $IP^{\star}$ (resp. $IP_{r}^{\star}$) if this set
intersects with every $IP$ set (resp. $IP_{r}$ sets). Note that every $IP_{r}$
set contains a $\varDelta_{r}$ set. To check this, let $FS\left(\langle
x_{n}\rangle_{n=1}^{r}\right)$ be an $IP_{r}$ set and let

$S=\left\\{x_{1},x_{1}+x_{2},\ldots,x_{1}+x_{2}+\cdots+x_{n}\right\\},$

Now $FS\left(\langle x_{n}\rangle_{n=1}^{r}\right)$ contains elements of the
form $\left\\{s-t:s>t\text{ and }s,t\in S\right\\}$. Hence every
$\varDelta_{r}^{\star}$ set is $IP_{r}^{\star}$. Again every $IP$ set contains
an $IP_{r}$ set for some $r\in\mathbb{N},$ hence every $IP_{r}^{\star}$ set is
$IP^{\star}$. Hence $\mathcal{K}$ is $IP_{721}^{\star}$ and hence an
$IP^{\star}$ set. For details on these sets the reader can see the book [9].
Let us recall the following theorem of N. Hindman [8]. Before that recall a
sub IP set of $FS\left(\langle x_{n}\rangle_{n=1}^{\infty}\right)$ is of the
form $FS\left(\langle y_{n}\rangle_{n=1}^{\infty}\right)\subseteq
FS\left(\langle x_{n}\rangle_{n=1}^{\infty}\right)$, where for each
$i\in\mathbb{N},$ $y_{i}=\sum_{t\in H_{i}}x_{t}$ and $H_{i}\cap
H_{j}=\emptyset$ for $i\neq j.$

######  Lemma 4.1.

[8, Hindman theorem] For every finite partition of an $IP$ set, there exists a
partition which contains a sub-IP set.

First we will prove three lemmas before we prove Theorem 1.5. The following
lemma says that dilation of any $IP^{\star}$ set by a number is again an
$IP^{\star}$ set.

######  Lemma 4.2.

Let $A\subseteq\mathbb{N},$ be an $IP^{\star}$ set. Then for any
$m\in\mathbb{N}$, $m\cdot A=\left\\{mx:x\in A\right\\}$ is again an
$IP^{\star}$ set.

###### Proof.

Let $\langle x_{n}\rangle_{n\in\mathbb{N}}$ be any sequence. For each
$i\in\mathbb{N}$, let

$x_{i}\equiv j\mod m$

where $j\in\left\\{0,1,\ldots,m-1\right\\}.$ Now pick $H_{1}$ (consider a
finite $H_{1}$ to be the collection of those $i$’s such that all $j$’s are
same) such that $m|\sum_{t\in H_{1}}x_{t}.$ Now continue this process to
obtain a disjoint sequences $\langle H_{n}\rangle_{n\in\mathbb{N}}$ of the
finite subsets of $\mathbb{N}$ such that $m|\sum_{t\in H_{n}}x_{t}$ for each
$n\in\mathbb{N}$. Now choose a new sequence $\langle
y_{n}\rangle_{n\in\mathbb{N}}$ such that $y_{n}=\frac{1}{m}\sum_{t\in
H_{n}}x_{t}$ for each $n\in\mathbb{N}$. Then $A\cap FS\left(\langle
y_{n}\rangle_{n\in\mathbb{N}}\right)\neq\emptyset$ and this implies $m\cdot
A\cap FS\left(\langle x_{n}\rangle_{n\in\mathbb{N}}\right)\neq\emptyset$,
finishing the proof.

The following lemma says that any $IP^{\star}$ set contains an $IP$ set. In
fact it contains a sub $IP$ set of any given $IP$ set.

######  Lemma 4.3.

Let $FS(\langle x_{n}\rangle_{n})$ be any $IP$ set and let $A$ be any
$IP^{\star}$ set. Then there exists a sub $IP$ set $FS(\langle
y_{n}\rangle_{n})$ of $FS(\langle x_{n}\rangle_{n})$ such that $A$ contains
$FS(\langle y_{n}\rangle_{n})$.

###### Proof.

Let us choose the following partition of $FS(\langle x_{n}\rangle_{n})$ by,

$FS(\langle x_{n}\rangle_{n})=(A\cap FS(\langle
x_{n}\rangle_{n}))\cup(FS(\langle x_{n}\rangle_{n})\setminus A).$

Now from Lemma 4.1, we have a sub-IP set $FS(\langle
y_{n}\rangle_{n}))\subseteq FS(\langle x_{n}\rangle_{n}))$ such that either
$FS(\langle y_{n}\rangle_{n}))\subseteq A\cap FS(\langle x_{n}\rangle_{n})$ or
$FS(\langle y_{n}\rangle_{n}))\subseteq FS(\langle x_{n}\rangle_{n})\setminus
A.$ But $A$ is an $IP^{\star}$ set, hence the second case is not possible. So,
$FS(\langle y_{n}\rangle_{n}))\subseteq A$, and the lemma follows.

We need the following lemma in our proof. This lemma says that if we take the
intersection of finitely many $IP^{\star}$ sets, then still it will be an
$IP^{\star}$ set.

######  Lemma 4.4.

Intersection of finitely many $IP^{\star}$ sets is again an $IP^{\star}$ set.

###### Proof.

Let $n\in\mathbb{N}$. Let $A_{1},A_{2},\ldots,A_{n}$ be $n$ many $IP^{\star}$
sets . Now assume that $FS(\langle x_{n}\rangle_{n})$ is an $IP$ set. From
Lemma 4.3, we have a sub $IP$ set $FS(\langle y_{n}\rangle_{n}))\subseteq
A_{1}\cap FS(\langle x_{n}\rangle_{n})$ and so $FS(\langle
y_{n}\rangle_{n}))\subseteq A_{1}.$ Now again applying Lemma 4.3, we have
$FS(\langle z_{n}\rangle_{n}))\subseteq A_{2}\cap FS(\langle
y_{n}\rangle_{n})$ and so $FS(\langle z_{n}\rangle_{n}))\subseteq A_{2}.$
Hence $FS(\langle z_{n}\rangle_{n}))\subseteq A_{1}\cap A_{2}.$ Iterating this
argument we have a sub $IP$ set $FS(\langle a_{n}\rangle_{n})$ of $FS(\langle
x_{n}\rangle_{n})$ such that $FS(\langle a_{n}\rangle_{n})\subseteq A_{1}\cap
A_{2}\cap\ldots\cap A_{n}.$ As $FS(\langle x_{n}\rangle_{n})$ is any $IP$ set,
arbitrarily chosen, we have $A_{1}\cap A_{2}\cap\ldots\cap A_{n}$ is an
$IP^{\star}$ set. This accomplishes the proof.

The following theorem is an abstract formulation of our Theorem 1.5. In fact
this result is much stronger.

######  Theorem 4.5.

If $A$ is any $IP^{\star}$ set and $B$ is an $IP$ set, then

$\mathbb{Q}_{>0}=\frac{A}{B}=\left\\{a/b:a\in A,b\in B\right\\}.$

###### Proof.

Let $\frac{m}{n}\in\mathbb{Q}_{>0}.$ Now $m\cdot B$ is an $IP$ set, and from
Lemma 4.2, we have $n\cdot A$ is an $IP^{\star}$ set. Hence, $n\cdot A\cap
m\cdot B\neq\emptyset.$ Let $x=na=mb$, where $a\in A,b\in B.$ Then $m/n=a/b\in
A/B.$ This completes the proof.

Now we are ready to prove our Theorem 1.5.

###### Proof of Theorem 1.5.

As $\mathcal{K}$ is an $IP_{721}^{\star}$ set, it is an $IP^{\star}$ set.
Again using Lemma 4.3, we can say that $\mathcal{K}$ contains an $IP$ set.
Hence from Theorem 4.5, we have our desired result.

The above proof of Theorem 1.5 shows a more powerful result, stated below.

######  Remark.

For any $IP$ set $\mathcal{D}\subset\mathcal{K}$, we have
$\mathbb{Q}_{>0}=\frac{\mathcal{K}}{\mathcal{D}}$, and hence also
$\mathbb{Q}_{>0}=\frac{\mathcal{D}}{\mathcal{K}}$.

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