Inverse questions for the large sieve

TL;DR

This paper investigates the inverse large sieve problem, exploring whether sets with certain residue class restrictions are essentially quadratic, providing partial results and conjectures but not a complete solution.

Contribution

It offers new results and conjectures on the inverse large sieve problem, especially regarding sets with additive structure, advancing understanding but not fully resolving the problem.

Findings

Established bounds for sets with residue class restrictions

Identified cases where bounds can be improved with additive structure

Formulated conjectures guiding future research

Abstract

Suppose that an infinite set $A$ occupies at most $\frac{1}{2}(p+1)$ 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.