arXiv subjects

George Shakan

Publications and source records attributed to George Shakan.

At least 19 recordsLinked to original sources

arXiv2023

Effective results on the size and structure of sumsets

Let $A \subset \mathbb{Z}^d$ be a finite set. It is known that $NA$ has a particular size ($\vert NA\vert = P_A(N)$ for some $P_A(X) \in \mathbb{Q}[X]$) and structure (all of the lattice points in a cone other than certain exceptional sets), once $N$ is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary $A$. Such explicit results were only previously known in the special cases when $d=1$, when the convex hull of $A$ is a simplex or when $\vert A\vert = d+2$, results which we improve.

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arXiv2022

A Generalization of the Schur-Siegel-Smyth Trace Problem

Let $α$ be a totally positive algebraic integer, and define its absolute trace to be $\frac{Tr(α)}{\text{deg}(α)}$, the trace of $α$ divided by the degree of $α$. Elementary considerations show that the absolute trace is always at least one, while it is plausible that for any $ε>0$, the absolute trace is at least $2-ε$ with only finitely many exceptions. This is known as the Schur-Siegel-Smyth trace problem. Our aim in this paper is to show that the Schur-Siegel-Smyth trace problem can be considered as a special case of a more general problem.

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arXiv2022

On the largest sum-free subset problem in the integers

Let $A \subset \mathbb{Z}_{>0}$ of size $n$. It is conjectured that for any $C >0$ and $n$ large enough that $A$ contains a sum-free subset of size at least $n/3 +C$. We study this problem and find an alternate proof of Bourgain's result that one make take $C=2/3$.

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arXiv2022

Explicit RIP matrices: an update

Leveraging recent advances in additive combinatorics, we exhibit explicit matrices satisfying the Restricted Isometry Property with better parameters. Namely, for $\varepsilon=3.26\cdot 10^{-7}$, large $k$ and $k^{2-\varepsilon} \le N\le k^{2+\varepsilon}$, we construct $n \times N$ RIP matrices of order $k$ with $k = Ω( n^{1/2+\varepsilon/4} )$.

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arXiv2021

Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants

We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree $n$ polynomials $f$ with $\mathrm{Gal}(f) \subseteq A_n$. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree $n$ monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree $n$ number fields with almost prime discriminants.

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arXiv2020

A large gap in a dilate of a set

Let $A \subset \mathbb{F}_p$ with $|A| > 1$. We show there is a $d \in \mathbb{F}_p^{\times}$ such that $d \cdot A$ contains a gap of size at least $2p/ |A| - 2 $.

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arXiv2020

The Frobenius postage stamp problem, and beyond

Let $A$ be a finite subset of $\mathbb{Z}^n$, which generates $\mathbb{Z}^n$ additively. We provide a precise description of the $N$-fold sumsets $NA$ for $N$ sufficiently large, with some explicit bounds on "sufficiently large."

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arXiv2020

An analytic approach to cardinalities of sumsets

Let $d$ be a positive integer and $U \subset \mathbb{Z}^d$ finite. We study $$β(U) : = \inf_{\substack{A , B \neq \emptyset \\ \text{finite}}} \frac{|A+B+U|}{|A|^{1/2}{|B|^{1/2}}},$$ and other related quantities. We employ tensorization, which is not available for the doubling constant, $|U+U|/|U|$. For instance, we show $$β(U) = |U|,$$ whenever $U$ is a subset of $\{0,1\}^d$. Our methods parallel those used for the Prékopa-Leindler inequality, an integral variant of the Brunn-Minkowski inequality.

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arXiv2020

A Weighted Prékopa-Leindler inequality and sumsets with quasicubes

We give a short, self-contained proof of two key results from a paper of four of the authors. The first is a kind of weighted discrete Prékopa-Leindler inequality. This is then applied to show that if $A, B \subseteq \mathbb{Z}^d$ are finite sets and $U$ is a subset of a "quasicube" then $|A + B + U| \geq |A|^{1/2} |B|^{1/2} |U|$. This result is a key ingredient in forthcoming work of the fifth author and Pälvölgyi on the sum-product phenomenon.

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arXiv2020

On generating functions in additive number theory, II: Lower-order terms and applications to PDEs

We obtain asymptotics for sums of the form $$ \sum_{n=1}^P e(α_kn^k + α_1n), $$ involving lower order main terms. As an application, we show that for almost all $α_2 \in [0,1)$ one has $$ \sup_{α_1 \in [0,1)} \Big| \sum_{1 \le n \le P} e(α_1(n^3+n) + α_2 n^3) \Big| \ll P^{3/4 + \varepsilon}, $$ and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrödinger and Airy equations.

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arXiv2019

On distinct consecutive differences

We show that if $A=\{a_1 < a_2 < \ldots < a_k\}$ is a set of real numbers such that the differences of the consecutive elements are distinct, then for and finite $B \subset \mathbb{R}$, $$|A+B|\gg |A|^{1/2}|B|.$$ The bound is tight up to the constant.

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arXiv2018

Stronger sum-product inequalities for small sets

Let $F$ be a field and a finite $A\subset F$ be sufficiently small in terms of the characteristic $p$ of $F$ if $p>0$. We strengthen the "threshold" sum-product inequality $$|AA|^3 |A\pm A|^2 \gg |A|^6\,,\;\;\;\;\mbox{hence} \;\; \;\;|AA|+|A+A|\gg |A|^{1+\frac{1}{5}},$$ due to Roche-Newton, Rudnev and Shkredov, to $$|AA|^5 |A\pm A|^4 \gg |A|^{11-o(1)}\,,\;\;\;\;\mbox{hence} \;\; \;\;|AA|+|A\pm A|\gg |A|^{1+\frac{2}{9}-o(1)},$$ as well as $$ |AA|^{36}|A-A|^{24} \gg |A|^{73-o(1)}. $$ The latter inequality is "threshold-breaking", for it shows for $ε>0$, one has $$|AA| \le |A|^{1+ε}\;\;\;\Rightarrow\;\;\; |A-A|\gg |A|^{\frac{3}{2}+c(ε)},$$ with $c(ε)>0$ if $ε$ is sufficiently small. This implies that regardless of $ε$, $$|AA-AA|\gg |A|^{\frac{3}{2}+\frac{1}{56}-o(1)}\,.$$

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arXiv2018

Fractal solutions of dispersive partial differential equations on the torus

We use exponential sums to study the fractal dimension of the graphs of solutions to linear dispersive PDE. Our techniques apply to Schrödinger, Airy, Boussinesq, the fractional Schrödinger, and the gravity and gravity-capillary water wave equations. We also discuss applications to certain nonlinear dispersive equations. In particular, we obtain bounds for the dimension of the graph of the solution to cubic nonlinear Schrödinger and Korteweg-de Vries equations along oblique lines in space-time.

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arXiv2018

On distinct consecutive $r$-differences

Suppose $A\subset \mathbb{R}$ of size $k$ has distinct consecutive $r$--differences, that is for $1 \leq i \leq k -r$, the $r$--tuples $$(a_{i+1} - a_i , \ldots , a_{i+r} - a_{i + r -1})$$ are distinct. Then for any finite $B \subset \mathbb{R}$, one has $$|A+B| \gg_r |A||B|^{1/(r+1)}.$$ Utilizing de Bruijn sequences, we show this inequality is sharp up to the constant. Moreover, for the sequence $\{nα\}$, a sharp upper bound for the size of the distinct consecutive $r$--differences is obtained, which generalizes Steinhaus' three gap theorem. A dual problem on the consecutive $r$--differences of the returning times for some $ϕ\in \mathbb{R}$ defined by $\{T : \{Tθ\}<ϕ\}$ is also considered, which generalizes a result of Slater.

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arXiv2018

On higher energy decompositions and the sum-product phenomenon

Let $A \subset \mathbb{R}$ be finite. We quantitatively improve the Balog-Wooley decomposition, that is $A$ can be partitioned into sets $B$ and $C$ such that $$\max\{E^+(B) , E^{\times}(C)\} \lesssim |A|^{3 - 7/26}, \ \ \max \{E^+(B,A) , E^{\times}(C, A) \}\lesssim |A|^{3 - 1/4}.$$ We use similar decompositions to improve upon various sum-product estimates. For instance, we show $$ |A+A| + |A A| \gtrsim |A|^{4/3 + 5/5277}.$$

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arXiv2018

Monochromatic Hilbert cubes and arithmetic progressions

The Van der Waerden number $W(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a monochromatic arithmetic progression of length $k$. Similarly, the Hilbert cube number $h(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a monochromatic affine $k$--cube, that is, a set of the form$$\left\{x_0 + \sum_{b \in B} b : B \subseteq A\right\}$$ for some $|A|=k$ and $x_0 \in \mathbb{Z}$. We show the following relation between the Hilbert cube number and the Van der Waerden number. Let $k \geq 3$ be an integer. Then for every $ε>0$, there is a $c > 0$ such that $$h(k,4) \ge \min\{W(\lfloor c k^2\rfloor, 2), 2^{k^{2.5-ε}}\}.$$ Thus we improve upon state of the art lower bounds for $h(k,4)$ conditional on $W(k,2)$ being significantly larger than $2^k$. In the other direction, this shows that the if the Hilbert cube number is close its state of the art lower bounds, then $W(k,2)$ is at most doubly exponential in $k$. We also show the optimal result that for any Sidon set $A \subset \mathbb{Z}$, one has $$\left|\left\{\sum_{b \in B} b : B \subseteq A\right\}\right| = Ω( |A|^3) .$$

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arXiv2016

A lower bound for the least prime in an arithmetic progression

Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek lower bounds for $P(k)$. In particular, we show that for almost every $k$ one has $P(k) \gg ϕ(k) \log k \log_2 k \log_4 k / \log_3 k,$ answering a question of Ford, Green, Konyangin, Maynard, and Tao. We rely on their recent work on large gaps between primes. Our main new idea is to use sieve weights to capture not only primes, but also small multiples of primes. We also give a heuristic which suggests that $\liminf_{k} \frac{P(k)}{ ϕ(k) \log^2 k} = 1.$

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arXiv2015

Sum of many dilates

We show that for any coprime integers $λ_1 , \ldots , λ_k$ and any finite $A \subset \mathbb{Z}$, one has $$|λ_1 \cdot A + \ldots + λ_k \cdot A| \geq (|λ_1| + \ldots + |λ_k|)|A|- C,$$ where $C$ only depends on $λ_1 , \ldots , λ_k$.

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