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Additive Correlation and the Inverse Problem for the Large Sieve (1706.06958v2)
Published 21 Jun 2017 in math.NT
Abstract: Let $A\subset [1,N]$ be a set of positive integers with $|A|\gg \sqrt N$. We show that if avoids about $p/2$ residue classes modulo $p$ for each prime $p$, the $A$ must correlate additively with the squares $S={n2:1\leq n\leq \sqrt N}$, in the sense that we have the additive energy estimate $E(A,S)\gg N\log N$. This is, in a sense, optimal.