Inverse questions for the large sieve

2013/11/24 by Ben J. Green, Green, Ben J., Adam J. Harper +1 · 3 citations
Mathematics · #Limits and Structures in Graph Theory #Analytic Number Theory Research #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1311.6176

Abstract

Suppose that an infinite set A occupies at most (1)/(2)(p+1) residue classes modulo p, for every sufficiently large prime p. The squares, or more generally the integer values of any quadratic, are an example of such a set. By the large sieve inequality the number of elements of A that are at most X is O(X^1/2), and the quadratic examples show that this is sharp. The simplest form of the inverse large sieve problem asks whether they are the only examples. We prove a variety of results and formulate various conjectures in connection with this problem, including several improvements of the large sieve bound when the residue classes occupied by A have some additive structure. Unfortunately we cannot solve the problem itself.

Cited by

• Larger sieve with height function and uniform bounds for integral points on curves over number fields
• An inverse theorem on sets with rich additive structure modulo primes
• Distribution of squares modulo a composite number

Related

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• On the large sieve with sparse sets of moduli
• The large sieve with sparse sets of moduli
• A weighted entropy approach for the quadratic inverse large sieve conjecture
• Polynomial values modulo primes on average and sharpness of the larger sieve