Lower Bounds for Expressions of Large Sieve Type

TL;DR

This paper proves that the large sieve method is essentially optimal for most exponential sums, confirming a longstanding conjecture by Erdős and Rényi.

Contribution

It establishes the near-optimality of the large sieve for almost all exponential sums, resolving a conjecture in the field.

Findings

Large sieve is optimal for almost all exponential sums

Confirmed Erdős and Rényi's conjecture

Provides bounds matching the conjectured optimality

Abstract

We show that the large sieve is optimal for almost all exponential sums, thus proving a conjecture by Erd\"os and Renyi.