On suprema of autoconvolutions with an application to Sidon sets

Cloninger, Alexander; Steinerberger, Stefan

doi:10.1090/proc/13690

On suprema of autoconvolutions with an application to Sidon sets
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by Alexander Cloninger and Stefan Steinerberger PDF

Proc. Amer. Math. Soc. 145 (2017), 3191-3200 Request permission

Abstract:

Let $f$ be a nonnegative function supported on $(-1/4, 1/4)$. We show that \[ \sup _{x \in \mathbb {R}}{\int _{\mathbb {R}}{f(t)f(x-t)dt}} \geq 1.28\left (\int _{-1/4}^{1/4}{f(x)dx} \right )^2,\] where 1.28 improves on a series of earlier results. The inequality arises naturally in additive combinatorics in the study of Sidon sets. We derive a relaxation of the problem that reduces to a finite number of cases and yields slightly stronger results. Our approach should be able to prove lower bounds that are arbitrary close to the sharp result. Currently, the bottleneck in our approach is runtime: new ideas might be able to significantly speed up the computation.

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