G. Shakan

activity

20052022

most citedOn higher energy decompositions and the sum-product phenomenon

29 citations · 64 across the 16 of their papers we have counts for

collaborators

22 papers

math.NT2022★ 2 cited

On the largest sum-free subset problem in the integers

George Shakan

Let $A \subset Z_{> 0}$ 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…

math.CO2021

Explicit RIP matrices: an update

Kevin Ford, Denka Kutzarova, George Shakan

Leveraging recent advances in additive combinatorics, we exhibit explicit matrices satisfying the Restricted Isometry Property with better parameters. Namely, for $\varepsilon=3.26…

math.NT2021★ 10 cited

Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants

Theresa C. Anderson, Ayla Gafni, Robert J. Lemke Oliver +3

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…

math.CO2021

Effective results on the size and structure of sumsets

Andrew Granville, George Shakan, Aled Walker

Let $A \subset Z^{d}$ be a finite set. It is known that $N A$ has a particular size ( $∣ N A ∣ = P_{A} (N)$ for some $P_{A} (X) \in Q [X]$ ) and structure (all of the l…

math.NT2020

A large gap in a dilate of a set

George Shakan

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

math.NT2020

The Frobenius postage stamp problem, and beyond

Andrew Granville, George Shakan

Let $A$ be a finite subset of $Z^{n}$ , which generates $Z^{n}$ additively. We provide a precise description of the $N$ -fold sumsets $N A$ for $N$ sufficiently large, w…

math.NT2020★ 2 cited

A Weighted Prékopa-Leindler inequality and sumsets with quasicubes

Ben Green, Dávid Matolcsi, Imre Ruzsa +2

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 app…

math.NT2020

An analytic approach to cardinalities of sumsets

Dávid Matolcsi, Imre Ruzsa, George Shakan +1

Let $d$ be a positive integer and $U \subset Z^{d}$ finite. We study $$β(U) : = \inf_{\substack{A , B \neq \emptyset \\ \text{finite}}} \frac{|A+B+U|}{|A|^{1/2}{|B|^{1/2}}},…

math.NT2020

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

Julia Brandes, Scott T. Parsell, Konstantinos Poulias +2

We obtain asymptotics for sums of the form $n = 1 \sum P e (α_{k} n^{k} + α_{1} n),$ involving lower order main terms. As an application, we show that for almost all $α_{2} \in [0, 1)$ one…

math.CO2019

On distinct consecutive differences

Imre Ruzsa, George Shakan, Jozsef Solymosi +1

We show that if $A = {a_{1} < a_{2} < \dots < a_{k}}$ is a set of real numbers such that the differences of the consecutive elements are distinct, then for and finite $B \subset \mathbb…

math.CO2018

Stronger sum-product inequalities for small sets

Misha Rudnev, George Shakan, Ilya Shkredov

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…

math.CO2018★ 7 cited

Breaking the 6/5 threshold for sums and products modulo a prime

G. Shakan, I. D. Shkredov

Let $A \subset F_{p}$ of size at most $p^{3/5}$ . We show $∣ A + A ∣ + ∣ AA ∣ ≳ ∣ A ∣^{6/5 + c},$ for $c = 4/305$ . Our main tools are the cartesian product point--line incide…

math.CO2018

Monochromatic Hilbert cubes and arithmetic progressions

József Balogh, Mikhail Lavrov, George Shakan +1

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,…

math.NT2018★ 29 cited

On higher energy decompositions and the sum-product phenomenon

George Shakan

Let $A \subset R$ 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^{\…

math.AP2018★ 1 cited

Fractal solutions of dispersive partial differential equations on the torus

Burak Erdoğan, George Shakan

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 Sc…

math.NT2017★ 6 cited

On distinct consecutive $r$ -differences

Junxian Li, George Shakan

Suppose $A \subset R$ 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 +…

math.NT2016

A lower bound for the least prime in an arithmetic progression

Junxian Li, Kyle Pratt, George Shakan

Fix $k$ a positive integer, and let $ℓ$ be coprime to $k$ . Let $p (k, ℓ)$ denote the smallest prime equivalent to $ℓ (mod k)$ , and set $P (k)$ to be the maximum of all the $…

math.NT2015

A Generalization of the Schur-Siegel-Smyth Trace Problem

Kyle Pratt, George Shakan, Alexandru Zaharescu

Let $α$ be a totally positive algebraic integer, and define its absolute trace to be $\frac{T r ( α )}{deg ( α )}$ , the trace of $α$ divided by the degree of $α$ . Elementary consid…

math.NT2014★ 2 cited

Sum of dilates in vector spaces

Antal Balog, George Shakan

Let $d \geq 2$ , $A \subset Z^{d}$ be finite and not contained in a translate of any hyperplane, and $q \in Z$ such that $∣ q ∣ > 1$ . We show $$|A+ q \cdot A| \geq (|q|…

math.NT2014★ 4 cited

Sum of many dilates

George Shakan

We show that for any coprime integers $λ_{1}, \dots, λ_{k}$ and any finite $A \subset Z$ , one has $$|λ_1 \cdot A + \ldots + λ_k \cdot A| \geq (|λ_1| + \ldots + |λ_k|)|A|- C…

math.NT2013

On the Sum of Dilations of a Set

Antal Balog, George Shakan

We show that for any relatively prime integers $1 \leq p < q$ and for any finite $A \subset Z$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$…

math.CO2005★ 1 cited

On distinct consecutive differences

J. Solymosi

We show that if $A = {a_{1}, a_{2}, ..., a_{k}}$ is a monotone increasing set of numbers, and the differences of the consecutive elements are all distinct, then $∣ A + B ∣ \geq c ∣ A ∣^{1/2} ∣ B ∣$ f…

On the largest sum-free subset problem in the integers

Explicit RIP matrices: an update

Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants

Effective results on the size and structure of sumsets

A large gap in a dilate of a set

The Frobenius postage stamp problem, and beyond

A Weighted Prékopa-Leindler inequality and sumsets with quasicubes

An analytic approach to cardinalities of sumsets

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

On distinct consecutive differences

Stronger sum-product inequalities for small sets

Breaking the 6/5 threshold for sums and products modulo a prime

Monochromatic Hilbert cubes and arithmetic progressions

On higher energy decompositions and the sum-product phenomenon

Fractal solutions of dispersive partial differential equations on the torus

On distinct consecutive r-differences

A lower bound for the least prime in an arithmetic progression

A Generalization of the Schur-Siegel-Smyth Trace Problem

Sum of dilates in vector spaces

Sum of many dilates

On the Sum of Dilations of a Set

On distinct consecutive differences

On distinct consecutive $r$ -differences