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.References
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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
- MR Author ID: 869041
- ORCID: 0000-0002-7745-4217
- Email: stefan.steinerberger@yale.edu
- Received by editor(s): April 23, 2016
- Published electronically: April 28, 2017
- Communicated by: Alexander Iosevich
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3191-3200
- MSC (2010): Primary 11B75, 42A85
- DOI: https://doi.org/10.1090/proc/13690
- MathSciNet review: 3652775