# Upper and lower bounds on the size of $B_{k}[g]$ sets

Griffin Johnston Department of Mathematics & Statistics, Villanova University,
Villanova, PA, U.S.A. $\\{$jjohns79<EMAIL_ADDRESS>The
second author is partially supported by National Science Foundation grant
DMS-2011553. Michael Tait11footnotemark: 1 Craig Timmons Department of
Mathematics and Statistics, California State University Sacramento, U.S.A.
<EMAIL_ADDRESS>Research is supported in part by Simons Foundation
Grant #359419

###### Abstract

A subset $A$ of the integers is a $B_{k}[g]$ set if the number of multisets
from $A$ that sum to any fixed integer is at most $g$. Let $F_{k,g}(n)$ denote
the maximum size of a $B_{k}[g]$ set in $\\{1,\dots,n\\}$. In this paper we
improve the best-known upper bounds on $F_{k,g}(n)$ for $g>1$ and $k$ large.
When $g=1$ we match the best upper bound of Green with an improved error term.
Additionally, we give a lower bound on $F_{k,g}(n)$ that matches a
construction of Lindström while removing one of the hypotheses.

##  1 Introduction

We will denote the integers $\\{1,2,\dots,n\\}$ by $[n]$. Given natural
numbers $k$ and $g$, a subset of $\mathbb{Z}$ is called a $B_{k}[g]$ set if
for any $m$ there are at most $g$ multisets $\\{x_{1},\dots,x_{k}\\}$ such
that $x_{1}+\cdots+x_{k}=m$ and $x_{i}\in A$. Determining bounds on the
maximum size of a $B_{k}[g]$ set in $[n]$ is a difficult and well-studied
problem and is the focus of this paper. Let $F_{k,g}(n)$ be the maximum size
of a $B_{k}[g]$ set in $[n]$.

When $k=2$ and $g=1$, $B_{2}[1]$ sets are called Sidon sets and have been
studied extensively since being introduced by Sidon [25] in the context of
Fourier series, and then studied further by Erdős and Turán [13] from a
combinatorial perspective. It is known [4, 18] that $F_{2,1}(n)\sim n^{1/2}$,
and determining whether or not $F_{2,1}(n)=n^{1/2}+O(1)$ is a 500 USD Erdős
problem [12]. Very recently the error term was improved by Balogh, Füredi, and
Roy [2] to $F_{2,1}(n)\leq n^{1/2}+0.998n^{1/4}$ for sufficiently large $n$.
This represents the first improvement upon the error term in over 50 years.

For other choices of $k$ and $g$, the asymptotic behavior of $F_{k,g}(n)$ has
not been determined, yet there are upper and lower bounds that give the order
of magnitude as a function of $n$. First we discuss upper bounds.

If $A$ is a $B_{k}[g]$ set contained in $[n]$, then each of the
$\binom{|A|+k-1}{k}$ multisets of size $k$ from $A$ determines a sum which is
an integer in $[kn]$. Each such integer can appear as a sum at most $g$ times
so $\binom{|A|+k-1}{k}\leq gkn$. This implies $F_{k,g}(n)\leq(gk!kn)^{1/k}$
which is known as the trivial bound. When $g=1$, a $B_{k}[g]$ set is often
called a $B_{k}$ set. Nontrivial bounds on the size of a $B_{k}$ set were
given by Jia [16] and Kolountzakis [17] for $k$ even, and by Chen [6] for $k$
odd. These bounds show that

$F_{k,1}(n)\leq\left(\left\lfloor\frac{k}{2}\right\rfloor!\left\lceil\frac{k}{2}\right\rceil!kn\right)^{1/k}+O_{k}(1).$
(1.1)

When $k$ is large, these bounds were improved by Green [15], who proved

$F_{k,1}(n)\leq\left(\left\lceil\frac{k}{2}\right\rceil!\left\lfloor\frac{k}{2}\right\rfloor!\sqrt{\frac{\pi
k}{2}}(1+\epsilon_{k})n\right)^{1/k}.$ (1.2)

It is noted (see page 379 of [15]) that $\epsilon_{k}$ can be taken to be
$O(k^{-1/8})$.

For $g>1$, Cilleruelo, Ruzsa, and Trujillo [9] improved the trivial bound by
proving

$F_{k,g}(n)\leq\left(\frac{k!kgn}{1+\cos^{k}(\pi/k)}\right)^{1/k}.$ (1.3)

Using an idea that the author’s attribute to Alon, Cilleruelo and Jiménez-
Urroz [7] showed

$F_{k,g}(n)\leq\left(\sqrt{3k}k!gn\right)^{1/k}.$ (1.4)

When $3\leq k\leq 6$, (1.3) is better and (1.4) is better for large $k$.
Currently the best general upper bound, proved by the third author [28], is

$F_{k,g}(n)\leq(1+o(1))\left(\frac{x_{k}k!kgn}{\pi}\right)^{1/k}.$ (1.5)

Here $x_{k}$ is the unique real number in $(0,\pi)$ that satisfies $\frac{\sin
x_{k}}{x_{k}}=\left(\frac{4}{3-\cos(\pi/k)}-1\right)^{k}$. In [28] it is shown
that this upper bound is better than both (1.3) and (1.4) but that
$\frac{x_{k}k!kg}{\pi}\to\sqrt{3k}k!g$ as $k\to\infty$.

Our first main theorem is an upper bound for large $k$ that improves (1.5) and
matches (1.2). Our theorem improves the error term in (1.2) from $O(k^{-1/8})$
[15] to $O(k^{-1/3})$.

######  Theorem 1.6.

Let $A\subset[n]$ be a $B_{k}[g]$ set.

  * (i)

When $g=1$

$|A|\leq\left(\left\lceil\frac{k}{2}\right\rceil!\left\lfloor\frac{k}{2}\right\rfloor!\sqrt{\frac{\pi
k}{2}}(1+O(k^{-1/3}))n\right)^{\frac{1}{k}},$

and

  * (ii)

when $g>1$

$|A|\leq\left(\sqrt{\frac{\pi k}{2}}k!g(1+O(k^{-1/3}))n\right)^{\frac{1}{k}}.$

The proof of Theorem 1.6 uses the Berry-Esseen theorem and is inspired by
recent work of Dubroff, Fox, and Xu [11] who used a similar technique applied
to the Erdős distinct subset sums problem.

Next we turn to lower bounds. Bose and Chowla [4] constructed a $B_{k}$ set of
size $q$ in $\mathbb{Z}_{q^{k}-1}$ for $q$ an odd prime power, and this
implies that $F_{k,1}(n)\geq(1-o(1))n^{1/k}$. Lindström [18] showed

$F_{k,g}(n)\geq(1-o(1))(gn)^{1/k},$ (1.7)

when $g=m^{k-1}$ for some integer $m$. Cilleruelo and Jiménez-Urroz [7] showed
that for any $\epsilon>0$, there exists a constant $g(k,\epsilon)$ so that for
$g>g(k,\epsilon)$

$F_{k,g}(n)\geq\left((1-\epsilon)\sqrt{\frac{\pi k}{6}}gn\right)^{1/k}.$ (1.8)

Our second main theorem matches the bound in (1.7) and removes the requirement
that $g=m^{k-1}$.

######  Theorem 1.9.

For any integers $k\geq 2$ and $g\geq 1$, we have
$F_{k,g}(n)\geq(1-o(1))(gn)^{1/k}$.

Before continuing the discussion, it is important to note that shortly after a
preprint of this article was made available, the authors were notified by
Carlos Trujillo that while not stated explicitly, Theorem 1.9 follows from
results of Caicedo, Gómez, and Trujillo [5]. The proof ideas are similar with
the notable difference that [5] works first in a general setting, and then
specializes to known $B_{h}$-sets. In our work we focus only on Bose-Chowla
$B_{h}$ sets and consequently, the proof of the lower bound
$F_{k,g}(n)\geq(1-o(1))(gn)^{1/k}$ is shorter. However, we recommend [5] for
details on this technique and how it can be applied to $B_{h}$ sets of Bose
and Chowla, Ruzsa, and Gómez and Trujillo. We leave in the details of the
proof of Theorem 1.9 for completeness and because it includes a density of
primes argument that gives an asymptotic lower bound for all $n$; the theorem
in [5] applies only to an infinite sequence in $n$.

Previous work on bounding $F_{k,g}(n)$ is extensive and we have not included
much of it. In particular, there are numerous papers that consider the case
when $k=2$ and $g>1$ (e.g. [8, 10, 17, 21, 20, 30, 29]). For more information,
see the surveys of Plagne [24] and O’Bryant [23]. In Section 2 we prove
Theorem 1.6 and in Section 3 we prove Theorem 1.9.

##  2 Upper bounds

For a finite set $S\subset\mathbb{N}$, we define
$\mathbb{E}(S)=\frac{1}{|S|}\sum_{s\in S}s$ and
$\mathrm{Var}(S)=\frac{1}{|S|}\sum_{s\in S}(s-\mathbb{E}(S))^{2}$. For any
random variable $X$ let $f_{X}$ be its probability distribution function and
let $F_{X}$ be its cumulative distribution function.

Assume that $A$ is a $B_{k}[g]$ set in $[n]$. To give an upper bound for the
size of $A$, we will consider the distribution of sums of random elements of
$A$. Define random variables $X_{i}$ that are independent and identically
distributed by

$\mathbb{P}(X_{i}=a-\mathbb{E}(A))=\frac{1}{|A|}$

for every $a\in A$. Note that $\mathbb{E}(X_{i})=0$ and
$\mathrm{Var}(X_{i})=\mathrm{Var}(A)$. Define $\delta$ so that
$\mathrm{Var}(A)=\delta n^{2}$. Any set of natural numbers up to $n$ has
variance at most $\frac{(n-1)^{2}}{4}$ and hence $\delta\leq 1/4$.

The details split into two cases depending on if $g=1$ or if $g>1$. When $g=1$
we take advantage of the fact that if $A$ is a $B_{k}[1]$ set, then for any
$c\in\mathbb{Z}$ there is at most one solution (up to rearranging) to the
equation

$a_{1}+\cdots+a_{\lceil k/2\rceil}-a_{\lceil k/2\rceil+1}-\cdots-a_{k}=c,$
(2.1)

where $a_{1},\cdots,a_{k}\in A$.

When $g>1$, there are at most $g$ solutions (up to rearranging) to the
equation

$a_{1}+\dots+a_{k}=c.$ (2.2)

Define

$\displaystyle Y_{1}$ $\displaystyle=X_{1}+\cdots+X_{\lceil k/2\rceil}$
$\displaystyle Y_{2}$ $\displaystyle=X_{{\lceil k/2\rceil}+1}+\cdots+X_{k}$
$\displaystyle Y$ $\displaystyle=Y_{1}-Y_{2}$ $\displaystyle Z$
$\displaystyle=X_{1}+\cdots+X_{k}.$

When $g=1$ we will consider the random variable $Y$ and when $g>1$ we will
consider the random variable $Z$.

In [7], Cilleruelo and Jiménez-Urroz give an upper bound on the size of a
$B_{k}[g]$ set for $g>1$ that depends on the variance of the set. Their proof
is easily modified to include the $g=1$ case, and for our purposes it is more
convenient to phrase the result in terms of the variance of the set. We give a
short proof for completeness.

######  Theorem 2.3 (Theorem 1.1 in [7]).

Let $k,g\in\mathbb{N}$ be fixed. If $A$ is a $B_{k}[g]$ set in $[n]$, then for
$g=1$ we have

$\frac{|A|^{2k}}{12\left(\lceil\frac{k}{2}\rceil!\lfloor\frac{k}{2}\rfloor!\right)^{2}}\leq(k+o(1))\mathrm{Var}(A),$

and for $g>1$ we have

$\frac{|A|^{2k}}{12(gk!)^{2}}\leq(k+o(1))\mathrm{Var}(A).$

###### Proof.

For convenience we assume that $|A|^{k}$ is divisible by
$\lceil\frac{k}{2}\rceil!\lfloor\frac{k}{2}\rfloor!$ when $g=1$ and by $gk!$
when $g>1$. If this is not the case we may truncate $A$ and let the $o(1)$
terms account for the difference.

The result follows by considering the variance of $Y$ when $g=1$ and the
variance of $Z$ when $g>1$. Because the $X_{i}$ are independent, we have that
$\mathrm{Var}(Y)=\mathrm{Var}(Z)=k\mathrm{Var}(A)$. To lower bound this
quantity, the variance of $Y$ or $Z$ is as small as possible when the values
taken by the random variables are as close together as possible. By (2.1) and
(2.2), when we look at all outputs of $Y$ or $Z$, each value can occur at most
$\lceil\frac{k}{2}\rceil!\lfloor\frac{k}{2}\rfloor!$ times for $g=1$ and at
most $gk!$ times for $g>1$.

Hence we have that the variance is bounded below by the variance of the
multiset of integers from $1$ to $\ell$ where
$\ell=\frac{|A|^{k}}{\lceil\frac{k}{2}\rceil!\lfloor\frac{k}{2}\rfloor!}$ when
$g=1$, and where $\ell=\frac{|A|^{k}}{gk!}$ when $g>1$. Since each integer
occurs the same number of times in this multiset, the variance of the multiset
is the same as that of the discrete uniform distribution of integers up to
$\ell$. This is given by

$\mathrm{Var}(\\{1,\cdots,\ell\\})=\frac{\ell^{2}-1}{12},$

and the result follows. ∎

When $k$ gets large, we can improve Theorem 2.3 by using more precise
information about $Y$ and $Z$ than the variance. As $k$ goes to infinity,
these distributions will be close to normal distributions, and we use the
Berry-Esseen theorem to quantify this.

######  Theorem 2.4 (Berry-Esseen Theorem).

Let $X_{1},...,X_{n}$ be independent random variables with
$\mathbb{E}[X_{i}]=0$, $\mathbb{E}[X_{i}^{2}]=\mathrm{Var}(X_{i})$ and
$\mathbb{E}[|X_{i}|^{3}]=\rho_{i}<\infty$. Let $X=X_{1}+\cdots+X_{n}$ and
$\sigma^{2}=\mathbb{E}[X^{2}]$. Then

$\sup_{x\in\mathbb{R}}|F_{X}(x)-\Phi(x)|\leq.56\psi,$

where $F_{X}(x)$ and $\Phi(x)$ are the cumulative distribution functions for
$X$ and the normal distribution with mean zero and standard deviation $\sigma$
respectively, and $\psi=\sigma^{-3}\cdot\sum_{i=1}^{n}\rho_{i}$.

By Theorem 2.4, for any $j$ we can approximate $X_{1}+\cdots+X_{j}$ by a
normal random variable with mean $0$ and variance $j\delta n^{2}$ by using

$\displaystyle\begin{split}\rho_{i}&=\mathbb{E}[|X_{i}|^{3}]\leq
n\mathbb{E}[X_{i}^{2}]=\delta n^{3},\\\
\sigma^{2}&=\mathrm{Var}(X)=j\mathrm{Var}(X_{i})=j\delta n^{2},\\\
\psi&=\frac{1}{\sigma^{3}}\sum_{i=1}^{j}\rho_{i}\leq\frac{j\delta
n^{3}}{j^{3/2}\delta^{3/2}n^{3}}=\frac{1}{\sqrt{j\delta}}.\end{split}$ (2.5)

We will approximate $Y_{1}$ by a normal distribution
$\mathcal{N}(0,\lceil{\frac{k}{2}\rceil}\delta n^{2})$ that has probability
distribution $\varphi_{1}(x)$ and cumulative distribution function
$\Phi_{1}(x)$. Similarly, let $\mathcal{N}(0,\lfloor\frac{k}{2}\rfloor\delta
n^{2})$ and $\mathcal{N}(0,k\delta n^{2})$ have probability distribution
functions $\varphi_{2}(x)$ and $\varphi(x)$ and cumulative distribution
functions $\Phi_{2}(x)$ and $\Phi(x)$, respectively. Since $F_{Y_{1}}$ and
$F_{Y_{2}}$ are close to $\Phi_{1}$ and $\Phi_{2}$ by Theorem 2.4, we have
that $F_{Y}$ is close to $\Phi$, quantified by the following lemma.

######  Lemma 2.6.

For $\Phi$ the cumulative distribution function of $\mathcal{N}(0,k\delta
n^{2})$, we have

$\sup_{x}|F_{Z}(x)-\Phi(x)|\leq\frac{.56}{\sqrt{k\delta}},$

and

$\sup_{x}\left|F_{Y}(x)-\Phi(x)\right|\leq
4\cdot\frac{.56}{\sqrt{\lfloor\frac{k}{2}\rfloor\delta}}.$

###### Proof.

The first inequality follows from Theorem 2.4 and (2.5). Now we prove the
second. Since $Y=Y_{1}-Y_{2}$ we have that $f_{Y}(x)=(f_{Y_{1}}\ast
f_{-Y_{2}})(x)$. We also have $\varphi=\varphi_{1}\ast\varphi_{2}$. Let $x$ be
arbitrary. Then

$\displaystyle|F_{Y}(x)-\Phi(x)|=$
$\displaystyle\left|\int_{-\infty}^{x}f_{Y_{1}}\ast
f_{-Y_{2}}-\varphi_{1}*\varphi_{2}\right|$ $\displaystyle=$
$\displaystyle\left|\int_{-\infty}^{x}\varphi_{2}\ast(f_{Y_{1}}-\varphi_{1})+\varphi_{1}\ast(f_{-Y_{2}}-\varphi_{2})+(f_{Y_{1}}-\varphi_{1})\ast(f_{-Y_{2}}-\varphi_{2})\right|$
$\displaystyle\leq$
$\displaystyle\left|\int_{-\infty}^{x}\varphi_{2}\ast(f_{Y_{1}}-\varphi_{1})\right|+\left|\int_{-\infty}^{x}\varphi_{1}\ast(f_{-Y_{2}}-\varphi_{2})\right|+\left|\int_{-\infty}^{x}(f_{Y_{1}}-\varphi_{1})\ast(f_{-Y_{2}}-\varphi_{2})\right|.$

We use Theorem 2.4 to show that each of the terms on the last line is small.

$\displaystyle\left|\int_{-\infty}^{x}\varphi_{2}\ast(f_{Y_{1}}-\varphi_{1})\right|$
$\displaystyle=\left|\int_{-\infty}^{x}\int_{-\infty}^{\infty}\varphi_{2}(z-y)\Big{(}f_{Y_{1}}(y)-\varphi_{1}(y)\Big{)}dzdy\right|$
$\displaystyle=\left|\int_{-\infty}^{x}f_{Y_{1}}(y)-\varphi_{1}(y)\
dy\int_{-\infty}^{\infty}\varphi_{2}(z-y)dz\right|$
$\displaystyle=\left|\int_{-\infty}^{x}f_{Y_{1}}(y)-\varphi_{1}(y)\
dy\right|\left|\int_{-\infty}^{\infty}\varphi_{2}(z-y)dz\right|$
$\displaystyle\leq\sup_{x\in\mathbb{R}}|F_{Y_{1}}(x)-\Phi_{1}(x)|\left|\int_{-\infty}^{\infty}\phi_{2}(z-y)dz\right|$
$\displaystyle\leq\frac{.56}{\sqrt{\delta\lceil\frac{k}{2}\rceil}}$

where the last inequality follows because $\varphi_{2}$ is a probability
distribution function and from (2.5). Similarly, noting that $\varphi_{2}$ is
symmetric around $0$, we have that

$\left|\int_{-\infty}^{x}\varphi_{1}\ast(f_{-Y_{2}}-\varphi_{2})\right|\leq\frac{.56}{\sqrt{\delta\lfloor\frac{k}{2}\rfloor}},$

and

$\left|\int_{-\infty}^{x}(f_{Y_{1}}-\varphi_{1})\ast(f_{-Y_{2}}-\varphi_{2})\right|\leq\left|\int_{-\infty}^{x}f_{Y_{1}}\ast(f_{-Y_{2}}-\varphi_{2})\right|+\left|\int_{-\infty}^{x}\varphi_{1}\ast(f_{-Y_{2}}-\varphi_{2})\right|\leq
2\frac{.56}{\sqrt{\delta\lfloor\frac{k}{2}\rfloor}}$

∎

We now have everything that we need to prove Theorem 1.6.

###### Proof of Theorem 1.6.

Let $k$ and $g$ be fixed and assume that $A$ is a $B_{k}[g]$ set in $[n]$. As
before, let $\mathrm{Var}(A)=\delta n^{2}$. If $\delta<\frac{\pi}{24}$, then
we may apply Theorem 2.3 to obtain the claimed upper bound. Hence for the
remainder of the proof we may assume that
$\frac{\pi}{24}\leq\delta\leq\frac{1}{4}$. Since the $X_{i}$ are independent
we have that the standard deviations of the random variables $Y$ and $Z$ are
the same and we will denote this quantity by $\sigma$. We will consider the
probability of the events that $-t<Y\leq t$ and $-t<Z\leq t$ where $t$ is an
integer that will be chosen later. For $g=1$, by Lemma 2.6 and the assumption
that $\delta\geq\pi/24$, we have that

$|F_{Y}(x)-\Phi(x)|\leq\frac{4\cdot
0.56}{\sqrt{\lfloor\frac{k}{2}\rfloor\frac{\pi}{24}}},$

for all $x$. Hence

$\displaystyle\mathbb{P}[-t<Y\leq t]$ $\displaystyle=F_{Y}(t)-F_{Y}(-t)$
$\displaystyle=\Phi(t)-\Phi(-t)-\Big{(}(F_{Y}(-t)-\Phi(-t))-(F_{Y}(t)-\Phi(t))\Big{)}$
$\displaystyle\geq\Phi(t)-\Phi(-t)-\Big{|}F_{Y}(-t)-\Phi(-t)\Big{|}-\Big{|}F_{Y}(t)-\Phi(t)\Big{|}$
$\displaystyle\geq\Phi(t)-\Phi(-t)-\frac{8\cdot
0.56}{\sqrt{\lfloor\frac{k}{2}\rfloor\frac{\pi}{24}}}$
$\displaystyle=\frac{1}{\sigma\sqrt{2\pi}}\int_{-t}^{t}\text{exp}(-\frac{x^{2}}{2\sigma^{2}})dx-\frac{4.48}{\sqrt{\lfloor\frac{k}{2}\rfloor\frac{\pi}{24}}}$
$\displaystyle\geq\frac{(2t)\cdot\exp{(-\frac{t^{2}}{2\sigma^{2}})}}{\sigma\sqrt{2\pi}}-\frac{4.48}{\sqrt{\lfloor\frac{k}{2}\rfloor\frac{\pi}{24}}}.$

On the other hand, by (2.1), we have that for any fixed $x$

$\mathbb{P}[Y=x]\leq\left(\left\lceil\frac{k}{2}\right\rceil!\right)\left(\left\lfloor\frac{k}{2}\right\rfloor!\right)|A|^{-k}.$

Combining these two inequalities yields

$\frac{(2t)\cdot\exp{(-\frac{t^{2}}{2\sigma^{2}})}}{\sigma\sqrt{2\pi}}-\frac{4.48}{\sqrt{\lfloor\frac{k}{2}\rfloor\frac{\pi}{24}}}\leq\mathbb{P}[-t<Y\leq
t]\leq(2t)\left(\left\lceil\frac{k}{2}\right\rceil!\right)\left(\left\lfloor\frac{k}{2}\right\rfloor!\right)|A|^{-k}.$

Using the inequality $1-x\leq e^{-x}$ for all $x$ and $\sigma^{2}\geq k\pi
n^{2}/24$, we have

$e^{\frac{-t^{2}}{2\sigma^{2}}}\geq 1-\frac{t^{2}}{2\sigma^{2}}\geq
1-\frac{12t^{2}}{\pi kn^{2}}.$

Applying this inequality, dividing both sides by $2t$ and using
$\sigma\leq\frac{\sqrt{k}n}{2}$ leads to

$\frac{1-\frac{12t^{2}}{\pi kn^{2}}}{n\sqrt{\pi
k/2}}-\frac{4.48}{2t\sqrt{\lfloor\frac{k}{2}\rfloor\frac{\pi}{24}}}\leq\left(\left\lceil\frac{k}{2}\right\rceil!\right)\left(\left\lfloor\frac{k}{2}\right\rfloor!\right)|A|^{-k}.$

Setting $t=k^{1/3}n$, we find that s

$\frac{1-\frac{12}{\pi k^{1/3}}}{n\sqrt{\pi
k/2}}-\frac{\frac{4.48}{2k^{1/3}n\sqrt{\lfloor\frac{k}{2}\rfloor\frac{\pi}{24}}}\cdot
n\sqrt{\pi k/2}}{n\sqrt{\pi
k/2}}\leq\left(\left\lceil\frac{k}{2}\right\rceil!\right)\left(\left\lfloor\frac{k}{2}\right\rfloor!\right)|A|^{-k}.$

Rearranging gives the result for $g=1$. For $g>1$ we use (2.2) and have that

$\mathbb{P}[Z=x]\leq gk!|A|^{-k},$

for any $x$. Performing a similar calculation on $\mathbb{P}(-k^{1/3}n<Z\leq
k^{1/3}n)$ gives the result, and we omit these details.

∎

##  3 Lower Bounds

In this section we prove Theorem 1.9. The idea is to begin with a known
construction of a $B_{k}[1]$ set and then consider the image of that set in a
quotient group. This idea has been used in other extremal graph theory and
combinatorial number theory problems before [1, 2, 14, 19, 22, 27]. In
particular, and as noted in the discussion following Theorem 1.9, Caicedo,
Gómez, and Trujillo [5] contains a more general approach than what is done
here.

Let $g$ and $k$ be fixed positive integers with $k\geq 2$. Assume that $q$ is
a power of prime such that $g$ divides $q-1$. Let $\mathbb{F}_{q^{k}}$ be the
finite field with $q^{k}$ elements and let $\theta$ be a generator of the
multiplicative group $\mathbb{F}_{q^{k}}^{*}$ of nonzero elements of
$\mathbb{F}_{q^{k}}$. Bose and Chowla [4] proved that

$A=\\{a\in\mathbb{Z}_{q^{k}-1}:\theta^{a}-\theta\in\mathbb{F}_{q}\\}$

is a $B_{k}[1]$ set in $\mathbb{Z}_{q^{k}-1}$. Let $\mu=\frac{q^{k}-1}{g}$ and
let $H$ be the subgroup of $\mathbb{Z}_{q^{k}-1}$ generated by $\mu$. Thus,
$H$ is the unique subgroup of $\mathbb{Z}_{q^{k}-1}$ with $g$ elements.

Next we prove a lemma that is essential to the construction. This lemma is
essentially known (Lemma 2.2 of [26] or Lemma 2.1 of [3]). A short proof is
included for completeness.

######  Lemma 3.1.

If $g$, $k$, $q$, $H$ and $A$ are given as above and $A-A:=\\{a-b:a,b\in
A\\}$, then $(A-A)\cap H=\\{0\\}$.

###### Proof.

Suppose $a,b\in A$ and $a-b\in H$. There is an element
$s\in\\{0,1,\dots,g-1\\}$ such that $a-b\equiv s\mu(\textup{mod}~{}q^{k}-1)$.
so $\theta^{a-b}=\theta^{s\mu}$ in $\mathbb{F}_{q^{k}}$. Let
$\alpha,\beta\in\mathbb{F}_{q}$ satisfy $\theta^{a}=\theta+\alpha$ and
$\theta^{b}=\theta+\beta$. Observe that
$(\theta^{s\mu})^{q-1}=(\theta^{q^{k}-1})^{\frac{s(q-1)}{g}}=1$ and so
$\theta^{s\mu}\in\mathbb{F}_{q}$. From $\theta^{a-b}=\theta^{s\mu}$, it
follows that $\theta+\alpha=\theta^{s\mu}(\theta+\beta)$ so

$(\theta^{s\mu}-1)\theta+\theta^{s\mu}\beta-\alpha=0.$

The minimal polynomial of $\theta$ over $\mathbb{F}_{q}$ has degree $k\geq 2$
and so we must have $\theta^{s\mu}-1=0$ and $\theta^{s\mu}\beta-\alpha=0$. In
particular, the first equation implies $s=0$ and so $a\equiv
b(\textup{mod}~{}q^{k}-1)$. ∎

In the quotient group $\Gamma:=\mathbb{Z}_{q^{k}-1}/H$, let $A_{H}=\\{a+H:a\in
A\\}$. If $a+H=b+H$ for some $a+H,b+H\in A_{H}$, then $a-b\in H$ which, by
Lemma 3.1, implies $a\equiv b(\textup{mod}~{}q^{k}-1)$. Hence,
$q=|A|=|A_{H}|$. Next, we prove that $A_{H}$ is a $B_{k}[g]$ set in $\Gamma$.
Suppose $c+H\in\Gamma$ and we have

$a_{1}+H+\dots+a_{k}+H=c+H$ (3.2)

for some $a_{i}+H\in A_{H}$. We will show that there are at most $g$ such
solutions up to the ordering of the terms on the left hand side of (3.2).
Indeed, (3.2) implies $a_{1}+\dots+a_{k}\equiv c+h(\textup{mod}~{}q^{k}-1)$
for some $h\in H$. There is at most one multiset $\\{a_{1},\dots a_{k}\\}$
from $A$ that is a solution to this equation. As there are $g$ choices for
$h$, there will be at most $g$ multisets $\\{a_{1}+H,\dots,a_{k}+H\\}$ from
$A_{H}$ that are solutions to (3.2). Therefore, $A_{H}$ is a $B_{k}[g]$ set in
$\Gamma$.

We now finish the proof using a density of primes argument. For positive
integers $x$, $c$, and $m$, let $\pi(x;c,m)$ be the number of primes $p$ for
which $p\leq x$ and $p\equiv c(\textup{mod}~{}m)$. Writing $\phi$ for the
Euler phi function, the prime number theorem in arithmetic progressions states
that if $\textup{gcd}(c,m)=1$, then

$\pi(x;c,m)=\frac{x}{\phi(m)\ln x}+O\left(\frac{x}{\ln^{2}x}\right).$

Let $\alpha=\lfloor(1-\epsilon)^{1/k}(gn)^{1/k}\rfloor$ and
$\beta=\lfloor(gn)^{1/k}\rfloor$. We then have

$\pi(\beta;1,g)-\pi(\alpha;1,g)\geq\frac{1}{\phi(g)}\left(\frac{\beta}{\ln\beta}-\frac{\alpha}{\ln\alpha}-O\left(\frac{\beta}{\ln^{2}\beta}\right)\right).$
(3.3)

For large enough $n$ depending on $\epsilon$, $g$, and $k$, the right hand
side of (3.3) is positive since $\frac{x}{\log x}$ is strictly increasing for
$x>e$. Thus, there is a prime $q$ with $q\equiv 1(\textup{mod}~{}g)$ and
$\alpha\leq q\leq\beta$. We can then choose a $B_{k}[g]$-set $A$ in the group
$\mathbb{Z}_{q^{k}-1}/H$ where $|A|=q$. This group is isomorphic to the cyclic
group $\mathbb{Z}_{(q^{k}-1)/g}$ and we let $A^{\prime}$ be a $B_{k}[g]$-set
in this cyclic group. Since $q\leq\beta$, we have $\frac{q^{k}-1}{g}\leq n$.
Therefore, we can view $A^{\prime}$ as a subset of $\\{1,2,\dots,n\\}$ and
under integer addition, $A^{\prime}$ is still a $B_{k}[g]$-set. It remains to
show that $q\geq(1-o(1))(gn)^{1/k}$, but this follows from the definition of
$\alpha$ and the inequality $q\geq\alpha$. We conclude that for all positive
integers $g$ and $k$ with $k\geq 2$,

$F_{k,g}(n)\geq(1-o(1))(gn)^{1/k}.$

## Acknowledgments

The authors would like to thank Carlos Trujillo for bringing [5] to our
attention.

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#  Derivation of the Deformed Heisenberg Algebra from Discrete Spacetime

Naveed Ahmad Shah<EMAIL_ADDRESS>Department of Physics, Jamia
Millia Islamia, New Delhi - 110025, India  Aasiya Shaikh
<EMAIL_ADDRESS>Design and Manufacturing Technology Division, Raja
Ramanna Centre for Advanced Technology (RRCAT), Indore-452013, Madhya Pradesh,
India.  Yas Yamin<EMAIL_ADDRESS>Irving K. Barber Faculty of Science,
University of British Columbia, 3333 University Way, Kelowna, British
Columbia, Canada V1V 1V7  P.K. Sahoo  Corresponding Author:
<EMAIL_ADDRESS>Department of Mathematics, Birla
Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.  Aaqid Bhat<EMAIL_ADDRESS>Department of Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.  Suhail Ahmad Lone
<EMAIL_ADDRESS>Department of Physics and Astrophysics, University
of Delhi, Delhi 110007, India  Mir Faizal<EMAIL_ADDRESS>Canadian Quantum Research Center 204-3002 32 Ave Vernon, BC V1T 2L7 Canada
Irving K. Barber Faculty of Science, University of British Columbia, 3333
University Way, Kelowna, British Columbia, Canada V1V 1V7  M. A. H. Ahsan
<EMAIL_ADDRESS>Department of Physics, Jamia Millia Islamia, New Delhi -
110025, India

###### Abstract

Even though the deformation of Heisenberg algebra by a minimal length has
become a main tool in quantum gravity phenomenology, it has never been
rigorously obtained and is derived using heuristic reasoning. Thus, for the
first time, we go beyond the heuristic derivation of deformed Heisenberg
algebra, and explicitly derive it using a model of discrete spacetime, which
will be motivated by quantum gravity. We first investigate the effects of
leading order Planckian lattice corrections, and demonstrate that they exactly
match those suggested by the heuristic arguments used in quantum gravity
phenomenology. However, as will rigorously obtain deformations from higher
order Planckian lattice corrections. Unlike the leading order corrections,
these higher order corrections will be model dependent. We will choose a
specific model, which will break the rotational symmetry, as it is important
to produce such effects as CMB anisotropies are thought to be related quantum
gravitational effects. We will propose based on the mathematical similarity of
the Planckian lattice used here with graphene, that graphene can be used as an
analogue system to study quantum gravity. Finally, we investigate the
deformation of the covariant form of the Heisenberg algebra using a four
dimensional Euclidean lattice.

##  I Introduction

There are strong indications that the Heisenberg algebra would be deformed due
to the quantum gravitational effects [1, 2, 3, 4]. Even though this
deformation has become the main tool used to study the quantum gravitational
modifications to various different physical systems [5, 6, 4], the deformation
itself is based on the heuristic arguments and has never been rigorously
derived [7, 8]. In constructing the heuristic argument the generalized
uncertainty principle is related to the existence of a minimal length, which
occurs in any approach to quantum gravity [5, 6, 4]. The existence of such a
minimal length can be is argued for using black hole physics, as the energy
needed to probe space at Planck scale is less than the energy needed to form a
mini black hole in that part of space [9, 10]. Hence, if we try to probe any
geometry below Planck scale, we will form a mini black hole, which in turn
will prevent such measurements from being made. Thus, it will not possible to
probe any length below Planck scale. Then it is argued that such a minimal
length would contradicts the usual uncertainty principle, where lengths can be
measured with arbitrary precision, if the momentum is not measured. To
reconcile quantum gravitational minimal length with foundations of quantum
mechanics, the usual uncertainty principle is modified to a generalized
uncertainty principle [7, 8]. Then it is argued that as the uncertainty
principle is closely related to the Heisenberg algebra, a modification of the
uncertainty principle will also modify the Heisenberg algebra [1, 2, 3, 4]. It
is this modified Heisenberg algebra, which has been used to study quantum
gravitational effects on various different physical systems [5, 6, 4].
However, this is derivation of the deformed Heisenberg algebra is very
heuristic, and this deformation has never been explicitly obtained in any
model of quantum gravity. Besides there is a fundamental problem with this
derivation, the minimal length can even occur in classical theories [11], so
it is not clear if it can have its origins in the uncertainty principle. It is
this point, along with a rigorous derivation of the deformation of the
Heisenberg algebra, that we will analyze.

Thus, we will go beyond the heuristic derivation of the deformed Heisenberg
algebra based on generalized uncertainty principle [7, 8], and obtain it
explicitly using discrete Planck scale spacetime lattice in quantum gravity.
We would like to point out that it is important to rigorously obtain the
deformation of the Heisenberg algebra from quantum gravity, as it can modify
the behavior of some important physical systems. It has been observed that the
deformation of the Heisenberg algebra removes the Chandrasekhar limit [12].
However, due to a cosmological constant the Chandrasekhar limit is restored
even with a deformed Heisenberg algebra [13]. Thus, the deformed Heisenberg
algebra can be related to the existence of cosmological constant [14]. It has
been argued that the modification of the black hole thermodynamics by the
deformation of the Heisenberg algebra would produce a black remnant [15, 16].
It has been proposed information of black hole can reside in such black
remnants, and this would be a possible resolution of the black hole
information paradox [17]. It has also been suggested that such a modification
of low energy effective field theory would produce a cutoff By placing a two-
level atom in a perfectly reflecting mirror, and analyzing its spontaneous
excitation using deformed Heisenberg algebra, it was observed that the
equivalence principle can be violated at short distances scales [18]. The
deformation of the Heisenberg algebra has been used to study the effect of
quantum gravitational corrections on Casimir effect in the two parallel
metallic plates [19]. The modification to quantum field theories from such a
deformation has also been studied [20]. In fact, explicit modification to both
abelian [21] and non-abelian [22] gauge theories from such deformation have
been investigated. Thus, it is physically interesting to study the deformation
of the Heisenberg algebra, which occurs due to a modification of the
uncertainty principle to a generalized uncertainty principle. However, the
only problem with this approach till date has been an absence of a rigorous
derivation of such a deformed algebra from any model of quantum gravity. This
is will be explicitly done in this paper using a specific model of quantum
gravity, where spacetime is discrete.

Various models of quantum gravity, such as causal set theory [23], loop
quantum gravity [24], multifractal spacetime [25], and spin foam [26] have
suggested that spacetime is discrete. Quantum graphity is a model of quantum
gravity, where spacetime is made up of where points are represented by nodes
on a graph, connected by links that can be on or off. This could in principle
analyzed as discrete lattice, with the nodes representing the lattice points.
Here, we associate a value with graph, and then physics is discussed at each
node. Each node represents a point, and the physics is discussed by analyzing
the interaction of this point with out points. As the quantum graphity has
been studied using honeycomb structure [27, 28]. At large distance the Planck
scale latter can be approximated using continuum field theories [29, 30]. The
mean field theory has been used to study the properties of spacetime described
by quantum graphity at low temperature [31]. It has also been observed in
quantum graphity, that Lorentz symmetry emerges as a low energy symmetry of
the theory, at large distances, and first order corrections to Lorentz
symmetry occur as first order corrections from quantum graphity [32]. The
honeycomb structure becomes important in models of dynamical triangulation in
quantum gravity [33, 34, 35, 36, 37]. In fact, we will demonstrate that an
advantage of the underlying trigonal symmetry of the honeycomb lattice is that
it can break rotation symmetry, and this could potentially explain the CMB
anisotropies [38, 39, 40, 41]. The honeycomb lattice has also been used in a
model of quantum gravity based on quantum Ricci curvature [42]. Thus, there
are strong motivations from various approaches to quantum gravity to assume
that spacetime can be represented honeycomb by a honeycomb discrete lattice,
rather than an Ising model [43, 44]. Here, we will use this model of discrete
spacetime to obtain the deformation of Heisenberg algebra.

It is important to note that the conclusions obtained here could also be hold
in theories like string theory. Even thought the spacetime is not discrete in
string theory, there is a limit to the scale at which the geometry can be
probed. This is because in string theory, the fundamental string is the
smallest probe to probe spacetime, and so it is not possible to probe geometry
below string length scale. Hence, the string length scale acts as a minimal
length scale in perturbative string theory [45]. Even through non-perturbative
effects in string theory produce point like objects like D0-branes, the
T-duality restricts limits the scale to which the theory can be probed [46].
It has been demonstrated that geometries where is there is a limit on the
scale at which the spacetime can be probed have the same information as
discrete spacetime [47]. Thus, even for string theory, it can be argued based
on the results from information theory [47], that discrete spacetime could
capture some important feature of string theory. Furthermore, it has been
demonstrated that spacetime geometries can emerge on boundaries of tensor
networks [48]. Here, it is interesting to note that holography has been used
to relate discrete spacetime described by tensor networks to string theory
[49, 50]. The honeycomb lattice has also been studied using holography in
string theory [51, 52]. It could thus be possible to holographically obtain
the deformation of the Heisenberg algebra. In this regard, the leading order
deformation of the Heisenberg algebra has been obtained using holography [53,
54]. However, this has not been done for next to the leading order
corrections, which will be different for theories with honeycomb lattice
structure [51, 52]. However, in the present work we will work directly with
models of discrete spacetime, and not take a holographic this approach to this
problem.

We will also comment on the analogue for such models of quantum gravity. It is
also possible to study the analogous quantum electrodynamics using graphene
[55, 56]. This is done by coupling the emergent Dirac equation with
electromagnetic potentials. In fact, various different effects predicted from
quantum electrodynamics have been tested using graphene [57, 58]. Similarly,
it has been demonstrated that non-abelian gauge fields can occur in graphene,
and so graphene can be used as an analogue system to study Yang-Mills theories
[59, 60]. Here, we go further and use of Dirac materials like graphene as an
analogue for quantum gravity [61, 62]. This is because in such materials, the
discrete lattice structure, it is not possible to probe graphene below
Planckian length scales, and so the Planckian length scales acts as a minimal
measurable length scale. This minimal measurable length scale deforms the
Heisenberg algebra used to define the Dirac structure in graphene [63, 64].
Here, we note that quantum electrodynamics has also been studied on discrete
spacetime lattice [65]. It has been argued that the underlying trigonal
symmetry of this honeycomb lattice can modify the low energy Heisenberg
algebra of graphene [66]. The underlying trigonal symmetry of the honeycomb
lattice has also been used to in investigating properties dynamical
triangulation in quantum gravity [33, 34, 35, 36, 37]. Thus, there are strong
motivations to use graphene as an analogue for quantum gravity.

As the deformation Heisenberg due to quantum gravity has only been obtained
using heuristic arguments [1, 2, 3, 4], we will make this heuristic argument
rigorous, and explicitly derive the deformation of the Heisenberg algebra
produced from discrete spacetime in quantum gravity. Finally, motivated by
such a deformation, we propose a very general deformation of covariant
Heisenberg algebra. To properly define and deform the covariant Heisenberg
algebra, we have to use a formalism of quantum mechanics, where time is viewed
as an quantum observable Thus, it is possible to consider time as a quantum
mechanical observable [67, 68, 69, 70, 71]. Here, we would like to point out
that by slightly modifying the foundations of quantum mechanics, it has been
possible to use time as a quantum observable. This has then been used to
define a minimal time, and a deformation of the commutator involving time [72,
53]. Here, we will generalize these results by considering a more general
deformation of the covariant algebra from quantum gravitational effects. The
specific forms of deformation, which in turn would depend on the details of
the quantum gravitational model, would produce different form of these
corrections. However, the important observation is that the leading order
corrections have a universal form, as the previously heuristic augmented has
suggested [1, 2, 3, 4]. Thus, the deformation of the Heisenberg has been used
to derive obtain important results in quantum gravity phenomenology. Here, we
will obtain such a deformation from a model of discrete spacetime.

##  II  Deformed Heisenberg Algebra

Discrete model like Ising model have been used in quantum gravity, where the
spacetime emerges from a Planckian Ising lattice [43, 44]. Apart from Ising
model, honeycomb structure for quantum gravity has been investigated in
quantum graphity [27, 28]. Here, we point that quantum graphity can be closely
linked to models of discrete spacetime. As in quantum graphity, points in
spacetime are represented by nodes on a graph, connected by links that can be
on or off. This can be mapped on to a model of discrete spacetime, again with
honeycomb lattice structure. Here, a lattice site can filled or empty. Thus,
there is almost a direct map between quantum graphity, and such a Planckian
lattice model of quantum gravity. A furthermore, advantage of taking a
honeycomb packing for discrete spacetime, rather than an Ising model [43, 44]
is that in this model it is naturally possible to account for the breaking of
isotropy of space, which could potentially explain the CMB anisotropies [38,
39, 40, 41]. This is due to the close relation between a Planckian honeycomb
lattice and dynamical triangulation [33, 34, 35, 36, 37]. The honeycomb
lattice has also been used in a model of quantum gravity where a quantum Ricci
curvature is obtained from discrete spacetime [42]. As the matter field in
nature are Fermionic, for quantum graphity with honeycomb lattice, we will
assume that the low energy effective field theory obtain from this lattice is
a Fermionic field theory. This is consistent with using graphene as an
analogue for this model of quantum gravity [61, 62]. Now for such a low energy
Fermionic field theory, we can write the dispersion relation using the analogy
of the dispersion relation in graphene. Now the the energy $E$ and as the
momentum $p$ of the fermionic field theory can obtained as the low energy
effective field theory from the Planckian lattice. Basically, as we have
assumed that spacetime can be obtained from a discrete Planckian lattice, we
can use the idea that same equations have same solutions, and this will hold
at any scale. This dispersion relation from an underlying honeycomb lattices
that have been studied for various systems [73, 74, 75, 76, 77, 78]. The
derivation of dispersion relation does not depend on the details of any
system, other than a underlying honeycomb lattice, and hence it will hold for
the honeycomb structure in quantum graphity [27, 28]. So, for the honeycomb
Planckian lattice, we can directly write the energy dispersion relation as
$E=(\lambda p-\alpha p^{2})$ [73, 74, 75, 76, 77, 78]. Here, in this Planckian
honeycomb lattice, we observe that in natural unites the Planck energy
$E_{P}=L_{P}^{-1}$ acts as an analogue for the lattice constant in Dirac
materials like graphene. Furthermore, the Fermi velocity is replaced by
velocity of light, which is set equal to one in natural unites, $c=1$. It is
due to this similarity of the modified dispersion relation [79, 80], that it
has been proposed that graphene can certainly be used as a quantum gravity
analogue [61, 62]. This similarity becomes explicit in models of quantum
gravity, with a Planck scale honeycomb discrete lattice. Here, we have defined
a deformation parameter, $\alpha$, which can be related to the the leading
order $T_{P}$ and the next to leading order $T^{\prime}_{P}$ correction terms
for the Planckian lattice. We also define $\lambda=\pm$, where its value
depends on if the lattice point is occupied or unoccupied. Then for any model,
with honeycomb lattice, we can write
$\alpha=({3}/{2}E_{P}){|T_{P}^{{}^{\prime}}|}/{T_{P}}$ [73, 74, 75, 76, 77,
78].

Now we will start our analysis using such an underlying discrete structure,
and then obtain the deformation of Heisenberg algebra using it. Using the
Dirac prescription $p\rightarrow\boldsymbol{\sigma}\cdot{\boldsymbol{p}}$ in
the expression for the energy, where ${\boldsymbol{p}}=-i\nabla$, is now the
momentum operator and $\boldsymbol{\sigma}$ are Pauli matrices, the low energy
effective Hamiltonian can be expressed as
${H}=\left[\boldsymbol{\sigma}\cdot{\boldsymbol{p}}-\alpha(\boldsymbol{\sigma}\cdot{\boldsymbol{p}})(\boldsymbol{\sigma}\cdot{\boldsymbol{p}})\right].$
The using this Hamiltonian, it is possible to define the generalized momentum
operator $\boldsymbol{{P}}={\boldsymbol{p}}(1-\alpha{p})$, such that
$\boldsymbol{\sigma}\cdot\boldsymbol{{P}}=\boldsymbol{\sigma}\cdot{\boldsymbol{p}}(1-\alpha\boldsymbol{\sigma}\cdot{\boldsymbol{p}})$.
Now the generalized momentum can be used to write the Hamiltonian as
${H}=\boldsymbol{\sigma}\cdot\boldsymbol{{P}}$. Even though the generalized
momentum ${P}_{i}$ are functions of ${p}_{i}$, we can choose
${X}_{i}={x}_{i}$, where ${X}_{i}$ is the generalized coordinate conjugate to
${P}_{i}$. As the low-energy coordinates and momentum operators satisfy the
standard commutation relations
$\left[{x}_{i},{x}_{j}\right]=\left[{p}_{i},{p}_{j}\right]=0$,
$~{}\left[{x}_{i},{p}_{j}\right]=i\delta_{ij}$, the generalized commutation
relations can be written as

$\displaystyle[{X}_{i},{P}_{j}]$ $\displaystyle=$ $\displaystyle
i\left[\delta_{ij}-\alpha\left({P}\delta_{ij}+\frac{{P}_{i}\
{P}_{j}}{{P}}\right)\right].$ (1)

The interesting observation here is that this deformation, which was obtained
rigorously from discrete spacetime, is exactly same as was heuristically
argued for using generalized uncertainty principle [1, 2]. Due to the
mathematical similarity of the underlying structure of discreteness between
this model of quantum gravity and graphene, it is expected that a similar
structure should also occur in graphene at leading order. In fact, a
generalized commutation has also been obtained for graphene by considering
next-to-nearest atomic interactions [63, 64], and even though it operates at a
different scale, it has the same form as the deformation obtained here. This
motivates the potential use of graphene as an analogue for such a model of
quantum gravity [61, 62]. Here, we have been able to directly make the
connection between this algebra and discrete spacetime using the analogy of
graphene with honeycomb lattice in quantum graphity. In fact, such an algebra
will hold for any lattice theory, where we consider both the nearest and next-
to-nearest lattice contributions to the low energy effective field theory.

Since the leading contribution of the high-energy momentum can be obtained by
neglecting the effect of nearby hopping terms, we can write it as a sum of the
low-energy momentum, and a correction term to it. So, we represent the
corrections to the low energy momentum as $\Delta p$, then we can express the
high energy momentum $\boldsymbol{p}$ as
$\boldsymbol{P}=\boldsymbol{p}+\Delta\boldsymbol{p}$. We can now obtain the
explicit expression for the correction term $\Delta p$ in terms of $p$ by
observing that $\Delta\boldsymbol{p}=(-\alpha\ p\ p_{x},-\alpha\ p\ p_{y}).$
Now because of the symmetric expansion (isotropic transformation on
$\boldsymbol{P}$) the rotational symmetry is preserved. The above
transformations allow consistent identification of the high-energy generalized
momentum as $P_{i}=p_{i}\left(1-a\ p\right),$ corresponding to the generalized
canonical coordinates $X_{i}$ [1, 2]. The full $SO(2,1)$ Lorentz symmetry is
broken by virtue of the presence of ${P_{i}P_{j}}/{P}$ term in the commutation
relations. Thus the modified Dirac equation corresponding to this modified
Dirac Hamiltonian is given by

$\left[-i\ \sigma_{i}\cdot\partial_{i}-\alpha\
\partial^{2}_{i}\right]\psi(\boldsymbol{r})=E\psi(\boldsymbol{r}).$ (2)

It is important to note that this is the modification to the Dirac equation
has been obtained from phenomenological considerations in quantum gravity [81,
82]. Here, it is obtained due to the leading order Planckian corrections. This
deformation indicates that this modification occurs in any system where a
discrete structure, if we consider the corrections beyond the continuum
approximation. The space is considered discrete in quantum gravity, and this
discrete space is approximated by continuum Dirac structure at the leading
order [81, 82]. The Planckian structure of quantum graphity is approximated by
a continuum Dirac structure at the leading order. Thus, by going beyond the
leading continuum approximation, we obtained a modification to the Dirac
equation. Now having studied the next-to-nearest hopping and leading order
contributions in quantum graphity, we can investigate the possible
modification due to quantum graphity, beyond leading order corrections.

##  III Rotational symmetry breaking

Figure 1: Orange: Energy plot for dispersion relation with leading order
correction terms and breaking rotational symmetry; Blue: Dispersion with
leading order correction terms without breaking rotational symmetry.

The advantage of using a rigorous derivation of the deformation is that we can
now precisely see how different models of quantum gravity can produce
different deformations of the Heisenberg algebra beyond the leading order. The
deformation we discussed in the previous section would be obtained from any
model of quantum gravity, as a universal deformation, and hence cannot
discriminate between different models of quantum gravity [5, 6, 4]. However,
going beyond that is not possible using the heuristic reasoning based on
generalized uncertainty principle. This can be done using the present rigorous
derivation, and we will now obtain the deformation, which explicitly breaks
the rotational symmetry of the effective field theory. This is important as
the deformation obtained in the previous section preserved the isotropy of
space due to its rotational symmetry. However, it is known that there are CMB
anisotropies [38, 39, 40, 41], and it has been suggested that they could be
explained using quantum gravitational corrections [83]. In fact, a deformation
of Heisenberg algebra, which breaks isotropy of space has been proposed to
address such phenomenological problems [84]. However, again this algebra has
been obtained using phenomenological considerations, without it being derived
from any model of quantum gravity. Here, we will obtain a deformation of the
Heisenberg algebra using the Planckian honeycomb lattice of quantum graphity.
It may be noted that in any honeycomb lattice there is an underlying trigonal
symmetry, as the lattice can be considered as composed of two interpenetrating
triangular lattices. This has important implications for graphene [66].
Furthermore, the trigonal symmetry in quantum gravity occur in models with
dynamical triangulation [33, 34, 35, 36, 37]. In fact, this approach is
basically based on a higher dimensional analogy of such a trigonal lattice.
Such a symmetry will also occur in quantum graphity with honeycomb lattice
[27, 28]. Thus, in models of discrete spacetime, like dynamical triangulation
[33, 34, 35, 36, 37], and quantum graphity with honeycomb lattice [27, 28], we
would expect deformation of low energy effective field theory. By analogy, we
also expect that this deformation in $(2+1)$ dimensions should resemble the
deformation produced by graphene [66] (except the scale of the deformation),
if we limit the discussion to spatial lattice in quantum gravity.

Now we observe from Fig (1), that even at relatively small energies, the
trigonal symmetry of the lattice non-trivially effects the energy dispersion.
We will again investigate the effects of such terms using third order in
expansion in the quasi-momentum $p$, without taking into consideration the
energy contributions from the next-to-nearest neighbor hopping. As we will see
the dispersion relation at this order furnishes us the trigonal warping terms
encoding the trigonal symmetry of the underlying quantum graphity lattice,
breaking the rotational symmetry. Thus, such terms could potentially be used
to explain CMB anisotropies [38, 39, 40, 41]. Now considering the general
rotational symmetry breaking expansion, which defines now the higher-energy
momentum $\boldsymbol{Q}_{0}$ in terms of low energy momentum $\boldsymbol{p}$
as; $\boldsymbol{Q}_{0}=\boldsymbol{p}+\Delta\boldsymbol{q}_{0}$, where we
have

$\Delta q_{0}=(\Delta q_{0,x},\Delta
q_{0,y})=\left(-\frac{1}{4E_{P}}(p^{2}_{x}-p^{2}_{y})-\frac{1}{8E^{2}_{p}}p_{x}(p^{2}_{x}+p^{2}_{y}),\
\frac{1}{2E_{P}}p_{x}p_{y}-\frac{1}{8E^{2}_{p}}p_{y}(p^{2}_{x}+p^{2}_{y})\right).$
(3)

Using the above expansion, the Hamiltonian upto
$\mathcal{O}((E_{p}^{-1}|p|)^{3})$ is given by

$\displaystyle H(p)$ $\displaystyle=$
$\displaystyle\Big{[}\sigma_{x}\left(p_{x}-\frac{1}{4E_{p}}(p^{2}_{x}-p^{2}_{y})-\frac{1}{8E^{2}_{p}}p_{x}(p^{2}_{x}+p^{2}_{y})\right)$
(4) $\displaystyle+$
$\displaystyle\sigma_{y}\left(p_{y}+\frac{1}{2E_{P}}p_{x}p_{y}-\frac{1}{8E_{p}^{2}}p_{y}(p^{2}_{x}+p^{2}_{y})\right)\Big{]}.$

Using this low energy effective Hamiltonian, we can derive the deformation of
the Heisenberg algebra produced by the third order correction terms. To obtain
such corrections to the Heisenberg algebra, we can define new generalized
momentum in terms of the standard low energy momentum. Thus, we define
$\boldsymbol{Q}_{0}$, with the two components as

$\displaystyle Q_{0,x}$ $\displaystyle=$
$\displaystyle\left(p_{x}-\frac{1}{4E_{p}}(p^{2}_{x}-p^{2}_{y})-\frac{1}{8E_{p}^{2}}p^{2}p_{x}\right),$
(5) $\displaystyle\ \ Q_{0,y}$ $\displaystyle=$
$\displaystyle\left(p_{y}+\frac{1}{2E_{p}}p_{x}p_{y}-\frac{1}{8E_{p}^{2}}p^{2}p_{y}\right).$
(6)

With the transformation, along with the identification of the corresponding
generalized coordinate with the standard low-energy ones
$X^{i}_{Q_{0}}=x^{i}$, we can write the Hamiltonian for the system in terms of
this new generalized momentum as
${H}=\boldsymbol{\sigma}\cdot\boldsymbol{Q}_{0}$, and particular form
deformation is realized as

$[x_{i},Q_{0,j}]=iF_{ij}(\boldsymbol{Q}_{0}),\ \
[x_{i},x_{j}]=0=[Q_{0,i},Q_{0,j}],$ (7)

where we have

$F_{ij}(\boldsymbol{Q}_{0})=\delta_{ij}+\frac{1}{2E_{p}}\begin{bmatrix}-Q_{0,x}&Q_{0,y}\\\
Q_{0,y}&Q_{0,x}\end{bmatrix}-\frac{1}{2E_{p}^{2}}\begin{bmatrix}Q^{2}_{0,x}&Q_{0,x}Q_{0,y}\\\
Q_{0,x}Q_{0,y}&Q^{2}_{0,y}\end{bmatrix}.$ (8)

Here again we observe that a similar algebra, which mathematically resembles
this algebra, but acts on a different scales, has been obtained for graphene
using tight-binding model [66]. This again points out to the potential use of
graphene as an analogue model for quantum gravity [61, 62]. The low energy
effective Hamiltonian, with next-to-nearest neighbor hopping up to third order
in momentum (which includes the trigonal warping term), can be written as

$\displaystyle H(p)$ $\displaystyle=$
$\displaystyle\Big{[}\sigma_{x}\left(p_{x}-\frac{1}{4E_{p}}(p^{2}_{x}-p^{2}_{y})-\frac{1}{8E_{p}^{2}}p_{x}(p^{2}_{x}+p^{2}_{y})\right)$
(9) $\displaystyle+$
$\displaystyle\sigma_{y}\left(p_{y}+\frac{1}{2E_{p}}p_{x}p_{y}-\frac{1}{8E_{p}^{2}}p_{y}(p^{2}_{x}+p^{2}_{y})\right)$
$\displaystyle-$
$\displaystyle\alpha\left(\frac{1}{E_{p}}(p^{2}_{x}+p^{2}_{y})-\frac{1}{2E_{p}^{2}}p^{3}_{x}+\frac{3}{2E_{p}^{2}}p_{x}p^{2}_{y}\right)\Big{]}.$

Now we can define a still more generalized momentum $\boldsymbol{Q}$ in terms
of the momentum $\boldsymbol{Q}_{0}$ as
$\boldsymbol{Q}_{i}=\boldsymbol{Q}_{0,i}(1-\alpha|\boldsymbol{Q}_{0}|)$.
Corresponding to the general momentum $\boldsymbol{Q}_{i}$,we now have
${X}_{i}$ as the generalized conjugate coordinate. Since ${x}_{i}$ and
${p}_{i}$ are the low-energy coordinates and momentum operators, they satisfy
the standard commutation relations, as discussed in the last section too. With
this generalized momentum a more generalized deformation of the form is
realized, given by

$\displaystyle[X_{i},Q_{j}]=iI_{ij}(\boldsymbol{Q}),\ \ \ \
[X_{i},X_{j}]=0=[Q_{i},Q_{j}],$ (10)

where the function $I_{ij}$ is given by

$I_{ij}(\boldsymbol{Q})=F_{ik}(\boldsymbol{Q})\left((1-\alpha|\boldsymbol{Q}|-2\alpha^{2}|\boldsymbol{Q}|^{2})\delta_{kj}-\alpha\frac{Q_{k}Q_{j}}{|\boldsymbol{Q}|}(1+\alpha|\boldsymbol{Q}|)\right).$
(11)

Here, $F_{ik}(\boldsymbol{Q})$ is further given by (8), where instead of
$\boldsymbol{Q}_{0}$ we will now have $\boldsymbol{Q}$. A similar algebra has
been obtained for graphene [85], due to the mathematical similarity of this
model with graphene. Hence, even to study such deformations, we can use
graphene as an analogue [61, 62]. In general, as we would like the deformation
to occur short distances, we expect the leading order contribution of this
algebra to coincide with the usual algebra. Thus, the most general form of
$I_{ij}(\boldsymbol{Q})$ can be written as

$I_{ij}(\boldsymbol{Q})=\delta_{ij}+\alpha f(\boldsymbol{Q})_{ij},$ (12)

where $f(\boldsymbol{Q})_{ij}$ is a suitable tensorial function formed from
the momentum ${Q}_{i}$. When we neglect the next-to-nearest hopping and only
consider the trigonal warping terms, it reduces to Eq. (8). Furthermore, by
considering both the trigonal warping and next-to-nearest hopping to third
order, we obtain Eq. (11). This algebra breaks the isotropy of space. The
breaking of isotropy has been motivated by quantum gravitational effects [84].
In string theory, isotropy can be broken on branes, and these anisotropic
branes are dual to a deformation of super-Yang-Mills theory by a position
dependent term [86, 87]. The isotropy can also be broken due to a
gravitational Higgs mechanism [88, 89]. Thus, it is important to generalize
this algebra to four dimensions, and use it to motivate a similar deformation
produced by quantum gravitational effects. However, before we generalize this
equation to four dimensions, we note that the specific form of
$I_{ij}(\boldsymbol{Q})$ depends on the symmetries of Planckian lattice
chosen. We would also like to point out that for the honeycomb Planckian
lattice, Dirac materials can be directly used as an analogue systems. This
makes it possible to use graphene as an analogue to study proposals used to
discuss CMB anisotropies [38, 39, 40, 41].

##  IV Covariant Deformation

In the previous sections, we explicitly analyzed the deformation of the
Heisenberg algebra from discrete spatial honeycomb lattice. To construct a
covariant generalization of such algebra, we need to first consistently define
the covariant algebra. The problem with defining a covariant algebra is that
time is not an operator in ordinary quantum mechanics. However, it can be
argued that time can be taken as a operator with slight modification to the
standard quantum mechanics. It is a widely accepted notion that time cannot be
expressed as a self-adjoint operator [90]. The reason for this is that a
Hamiltonian with a semi-bounded spectrum does not admit a group of shifts
which can be generated from canonically conjugate self-adjoint operators. This
limitation can be resolved in the von-Neumann formulation of quantum
mechanics. The resolution is based on the observation that quantum mechanics
need not be restricted to self-adjoint operators [91]. In fact, the momentum
operator for a free particle bounded by a rigid wall at (like time) is also
not a self-adjoint operator. It is possible to include such cases if quantum
mechanics is based on maximal Hermitian operator, and this is done in the von-
Neumann formation of quantum mechanics.

Thus, it is possible to consider time as a quantum mechanical observable [67,
68, 69, 70, 71]. It has also been proposed that time can be investigated using
its symmetric non-self-adjoint operators which satisfy [92, 93]

$[t,H]=-i.$ (13)

Here, observables are are positive operator valued measures. If $H$ is the
Hamiltonian of a system, then a unitary representation of the time translation
group can be written as $\tau\to e^{iH\tau}$. A quantum mechanical time
observable of the system is represented by the positive operator valued $B$,
such that $t\to B(t)$. Here, we have $e^{iH\tau}B(t)e^{-iH\tau}=B(t-\tau)$.
So, a symmetric time operator can be defined for a time observable $B$, such
that

$t=\int tdB(t).$ (14)

The probability measure $t\to p(t)$ in any experiment can be expressed as

$p(t)=tr[\rho B(t)].$ (15)

where $\rho$ in the density matrix of a state, and $t\to B(t)$ is a positive
operator valued measure [67]. Thus, the time can be represented as an
observable in quantum mechanics.

We will use this definition of time to define and deform the covariant
commutators. In fact, such a deformation of the commutator which corresponded
to a minimum measurable time interval can be expressed as [72, 53]. Here, the
observable time is defined with reference to the evolution of a non-stationary
quantity. The events were characterized by specific values of this quantity
[67], and then minimal value of time is defined with respect to the specific
values of this quantity. For example, for tunneling time for particles, a
minimum measurable time interval could be viewed a lower bound on the
measurement of tunneling time for particles [67, 68, 69, 70, 71]. This
deformed covariant algebra [72, 53], with a minimal time has been related to
time crystals, which is the temporal analogy of spatial crystals [94, 95].

It is possible to define a spacetime lattice structure, and this can be done
by wick rotating time in the complex plane $\tau=-it$ [96]. Now we can assume
that any continuum field theory on four dimensional Euclidean structure is a
low energy approximation to a discrete model of spacetime, with wick rotated
time. Motivated by the previous sections, we can also propose that this
Euclidean spacetime is represented by a four dimensional generalization of the
honeycomb lattice. Thus, we propose that a full deformation of a covariant
Heisenberg algebra can be written as

$\displaystyle[X_{\mu},K_{\nu}]=iI(\boldsymbol{K})_{\mu\nu},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\
[R_{\mu},R_{\nu}]=0=[K_{\mu},K_{\nu}],$ (16)

where $K_{\mu}$ is the four momentum, and $I(\boldsymbol{K})_{\mu\nu}$ is
given by

$I(\boldsymbol{K})_{\mu\nu}=\delta_{\mu\nu}+\alpha
f(\boldsymbol{K})_{\mu\nu}.$ (17)

Here, $f(\boldsymbol{K})_{\mu\nu}$ is a suitable tensorial function formed
from the four momentum ${K}_{\mu}$, such that it respects the symmetries of
the Planck scale lattice structure produced by quantum gravity. For example,
if we consider dynamical triangulation or quantum graphity with honeycomb
lattice, and neglect the temporal deformation of $(2+1)$ dimensions gravity,
then this tensorial function in Eq. (16) resembles the tensorial function of
Eq. (11), with the parameter $\alpha$ being of order of Planck length. Now for
quantum graphity with four dimensional covariant honeycomb lattice and
dynamical triangulation in four dimensions, we would expect a deformation in
Eq. Eq. (16), which would break the Lorentz symmetry, like the breaking of
isotropy of space by the deformation produced by third order correction in Eq.
(11). In the limit, higher order correction terms, like the trigonal warping
terms are neglected, and only the next-to-nearest hopping is considered, the
tensorial function $f(\boldsymbol{K})_{\mu\nu}$ can be expressed as

$f(\boldsymbol{K})_{\mu\nu}=\left({K}\delta_{\mu\nu}+\frac{{K}_{\mu}\
{K}_{\nu}}{{K}}\right).$ (18)

This deformation of the low energy Heisenberg algebra will occur due to most
four dimensional lattice structures. Hence, this deformation has been
motivated by the general consideration using quantum gravitational
phenomenology [72, 53]. However, the deformation produced by higher order
correction terms, will depend on the symmetry of the underlying theory. It is
possible to study different deformations of the covariant Heisenberg algebra,
which can correspond to low energy limits of different quantum theories of
gravity. Here, it is important to note that any symmetry like Lorentz symmetry
will break due to higher order Planckian lattice corrections, and these
symmetries will be restored if these higher order Planckian lattice
corrections are neglected. As these corrections become important near Planck
energy, at low energies comparable to Planck energy, when we neglect all
higher order corrections, we obtain the original covariant Heisenberg algebra.
Such breaking of Lorentz symmetry due to quantum gravitational effects have
already been studied in quantum gravity phenomenological [97, 98, 99, 100].
Here, we have investigated the reason for such breaking using a four
dimensional Euclidean Planckian lattice.

##  V Conclusion

The main result of this letter is that we have for the first time explicitly
obtained the deformation of the Heisenberg using a model of quantum gravity.
We observed that the leading order deformation is exactly the same as
suggested previously using heuristic arguments based on the generalized
uncertainty principle. To obtain these results, we have analyzed low energy
consequences of quantum graphity with Planck scale honeycomb lattice. As the
matter fields are fermionic, we assumed that the low energy effective field
theory of this lattice is also a fermionic field theory. We first analyzed the
effects of leading order corrections on this discrete Planckian lattice. It
was observed that the modifications produced by these are similar to the the
modifications proposed using quantum gravitational phenomenology. They also
resembled the modifications to the Dirac equation in graphene from the leading
order correction terms, and as such graphene can be used as an analogue system
to study it [61, 62]. Next, we investigated the effects of higher order
correction terms on this discrete Planckian lattice. We analyze this Planckian
lattice up to third order, which break the isotropy of space. We obtained a
general deformation of the Heisenberg algebra, which reduced the specific
forms of the algebra in different limits. To deformed the covariant algebra,
we used a formalism of quantum mechanics, where time could be taken as an
observable. The minimal time restricted the measurement of this observable. We
also analyzed various deformation of this covariant algebra.

There have been several attempts to detect the deformation of Heisenberg
algebra by Planck scale quantum gravitational effects. These include proposals
based on optomechanical setups [5, 6]. In this setup it was possible to study
the canonical commutation relation of the center of mass mode of a massive
mechanical oscillator, whose mass was almost of the order of Planck mass. The
quantum optical control and readout of the mechanical system was used to study
the effects of modification of Heisenberg algebra. It was also suggested that
the modification of Heisenberg algebra can be detected using ultra-precise
measurements of Lamb shift and Landau levels [4]. However, it is hard to
detect the Planck scale deformations directly. Here, we observe that models of
quantum gravity like quantum graphity with honeycomb lattice are similar to
graphene, except for the scale of the lattice structure. Thus, graphene can be
used as an analogue for quantum gravity [61, 62]. It can be used to study and
test quantum gravitational effects, due to the similar mathematical structure
between graphene and quantum gravitational systems. It may be noted that
analogue physics has produced important results for real systems. For example,
using black hole analogues, it was demonstrated that Hawking radiation is a
real thermal radiation [101]. Thus, important quantum gravitational phenomena
can be understood using graphene as its analogue.

The results of this letter were obtained by using a honeycomb structure on a
Planckian lattice. Due to the close analogy of this model of quantum gravity
with graphene, and as graphene can act as an analogue for this model of
quantum gravity [61, 62]. In fact, the results of this letter can have
directly application in graphene, due to similarly of the lattice structure
with quantum graphity and graphene. The algebra obtained in this paper will
hold for graphene, if we replace $E_{p}$ with the lattice constant for
graphene $a^{-1}$, and the velocity of light with the Fermi velocity. Thus,
the results of this paper can have important application for studding
condensed matter phenomena in graphene too. For several decades, graphene has
drawn the interest of numerous scientists due to its distinct composition of a
single layer of carbon atoms bonded in a hexagonal lattice pattern.
Additionally, materials based on graphene exhibit exceptional physical and
chemical characteristics, including a high surface area ($2630m^{2}/g$) [102,
103], excellent electrical conductivity ($1738siemens/m$) [102, 104], strong
mechanical strength (about $1100GPa$) [102], unparalleled thermal conductivity
($5000W/m/K$) [102], ultra-high charge carrier mobility and a linear
dispersion band structure [105]. Due to these unique properties, graphene is
considered a promising subject of research in several fields, including
biomedical [102] and biotechnology [106], chemistry and catalysis [107],
energy technology and environment [108], nanoscience [109, 105]. These have
been studied using linear approximation to the tight-binding model in
graphene, and by going beyond linear approximation, we improve all these
applications of graphene.

###### Acknowledgements.

We would like to thank Quach James for discussion related to quantum graphity.
We would also like to thank Salman Sajad Wani, Alonso Contreras Astorga and
François Fillion-Gourdeau for useful discussion about graphene.

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