Generalized difference sets and autocorrelation integrals
Tom 199 / 2021
Streszczenie
In 2010, Cilleruelo, Ruzsa, and Vinuesa established a surprising connection between the maximum possible size of a generalized Sidon set in the first 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.