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A large gap in a dilate of a set
George Shakan
Let with . We show there is a such that contains a gap of size at least .
The Frobenius postage stamp problem, and beyond
Andrew Granville, George Shakan
Let be a finite subset of , which generates additively. We provide a precise description of the -fold sumsets for sufficiently large, w…
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…
An analytic approach to cardinalities of sumsets
Dávid Matolcsi, Imre Ruzsa, George Shakan +1
Let be a positive integer and finite. We study $$β(U) : = \inf_{\substack{A , B \neq \emptyset \\ \text{finite}}} \frac{|A+B+U|}{|A|^{1/2}{|B|^{1/2}}},…
On generating functions in additive number theory, II: Lower-order terms and applications to PDEs
Julia Brandes, Scott T. Parsell, Konstantinos Poulias +2
We obtain asymptotics for sums of the form involving lower order main terms. As an application, we show that for almost all one…
On higher energy decompositions and the sum-product phenomenon
George Shakan
Let be finite. We quantitatively improve the Balog-Wooley decomposition, that is can be partitioned into sets and such that $$\max\{E^+(B) , E^{\…