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Additive correlation and the inverse problem for the large sieve



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.



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[BC] Bose, R. C. and Chowla, S. Theorems in the additive theory of numbers. Comment. Math. Helv. 37 (1962-63), 141147.
[CL] Croot, E. S. III, and Lev, V. F. Open problems in additive combinatorics. In Additive combinatorics, volume 43 of CRM Proc. Lecture Notes, pages 207–233 (Amer. Math. Soc., Providence, RI, 2007).
[FI] Friedlander, J. and Iwaniec, H. Opera de cribro. American Mathematical Society Colloquium Publications, 57 (American Mathematical Society, Providence, RI, 2010).
[GH] Green, B. and Harper, A. J Inverse questions for the large sieve. Geom. Funct. Anal. 24 (4) (2014), 11671203.
[HV] Helfgott, H. A. and Venkatesh, A. How small must ill-distributed sets be? Analytic Number Theory, Essays in honour of Klaus Roth (Cambridge University Press 2009), 224234.
[K] Kolountzakis, M. N. On the uniform distribution in residue classes of dense sets of integers with distinct sums. J. Number Theory 76 (1999), 147153.
[La] Landau, E. Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate. Arch. Math. Phys. 13 (1908), 305312.
[Li] Lindström, B. Well distribution of Sidon sets in residue classes. J. Number Theory 69 (1998), 197200.
[M] Montgomery, H. L. A note on the large sieve. J. London Math. Soc. 43 (1968), 9398.
[R] Ramanujan, S. Some formulae in the analytic theory of numbers. Messenger Math. 45 (1916), 8184.
[W1] Walsh, M. N. The inverse sieve problem in high dimensions. Duke Math. J. 161 (2012), no. 10, 20012022.
[W2] Walsh, M. N. The algebraicity of ill-distributed sets. Geom. Funct. Anal. 24 (2014), no. 3, 959967.

Additive correlation and the inverse problem for the large sieve



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