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Additive correlation and the inverse problem for the large sieve

Published online by Cambridge University Press:  09 July 2018

BRANDON HANSON
BRANDON HANSON*
Affiliation:
Dept. of Mathematics, Boyd Graduate Studies Research Center, University of Georgia, Athens, GA 30602, USA. e-mail: brandon.w.hanson@gmail.com
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Abstract

Let A ⊆ [1, N] be a set of integers with |A| ≫ $\sqrt N$. We show that if A avoids about p/2 residue classes modulo p for each prime p, then A must correlate additively with the squares S = {n2 : 1 ≤ n ≤ $\sqrt N$}, in the sense that we have the additive energy estimate

$$ E(A,S)\gg N\log N. $$
This is, in a sense, optimal.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 2 , March 2020 , pp. 211 - 217
Copyright
Copyright © Cambridge Philosophical Society 2018 

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References

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