G. Shakan

activity

20052020

most citedOn distinct consecutive differences

1 citations · 1 across the 3 of their papers we have counts for

collaborators

12 papers

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

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…