2020/04/14 by Noah Kravitz, Kravitz, Noah
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.2004.06611
arxiv created 2020/04/14 · arxiv updated 2020/04/15
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 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.