On Suprema of Autoconvolutions with an Application to Sidon sets

2014/03/31 by Alexander Cloninger, Stefan Steinerberger, Cloninger, Alexander +1 · 6 citations
Mathematics · #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CA #math.CO #math.NT

paper · pdf · doi:10.48550/arxiv.1403.7988

arxiv created 2016/04/22 · arxiv updated 2016/04/26

Abstract

Let f be a nonnegative function supported on (-1/4, 1/4). We show sup_{x ∈ ℝ}∫_ℝf(t)f(x-t)dt ≥ 1.28(∫_-1/4^1/4f(x)dx )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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