Generalized difference sets and autocorrelation integrals

Abstract

In 2010, Cilleruelo, Ruzsa, and Vinuesa established a surprising connection between the maximum possible size of a generalized Sidon set in the first $N$ natural numbers and the optimal constant in an ``analogous'' problem concerning nonnegative-valued functions on $[0,1]$ with autoconvolution integral uniformly bounded above. Answering a recent question of Barnard and Steinerberger, we prove the corresponding dual result about the minimum size of a so-called generalized difference set that covers the first $N$ natural numbers and the optimal constant in an analogous problem concerning nonnegative-valued functions on $\mathbb{R}$ with autocorrelation integral bounded below on $[0,1]$. These results show that the correspondence of Cilleruelo, Ruzsa, and Vinuesa is representative of a more general phenomenon relating discrete problems in additive combinatorics to questions in the continuous world.