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On suprema of autoconvolutions with an application to Sidon sets


Authors: Alexander Cloninger and Stefan Steinerberger
Journal: Proc. Amer. Math. Soc. 145 (2017), 3191-3200
MSC (2010): Primary 11B75, 42A85
DOI: https://doi.org/10.1090/proc/13690
Published electronically: April 28, 2017
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Abstract: Let $ f$ be a nonnegative function supported on $ (-1/4, 1/4)$. We show that

$\displaystyle \sup _{x \in \mathbb{R}}{\int _{\mathbb{R}}{f(t)f(x-t)dt}} \geq 1.28\left (\int _{-1/4}^{1/4}{f(x)dx} \right )^2,$

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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Additional Information

Alexander Cloninger
Affiliation: Department of Mathematics, Program in Applied Mathematics, Yale University, New Haven, Connecticut 06510
Email: alexander.cloninger@yale.edu

Stefan Steinerberger
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
Email: stefan.steinerberger@yale.edu

DOI: https://doi.org/10.1090/proc/13690
Keywords: Sidon sets, autoconvolutions.
Received by editor(s): April 23, 2016
Published electronically: April 28, 2017
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2017 American Mathematical Society