Additive energy and a large sieve inequality for sparse sequences

TL;DR

This paper develops a new large sieve inequality for sparse sequences based on additive energy, improving bounds for polynomial and special sequences, and applies it to enhance the Bombieri--Vinogradov theorem for Piatetski-Shapiro moduli.

Contribution

It introduces a large sieve inequality depending on additive energy for sparse sequences, extending previous bounds and applying to various polynomial and convex sequences.

Findings

Improved bounds for monomial moduli in large sieve inequalities.

Enhanced level of distribution in Bombieri--Vinogradov theorem for Piatetski-Shapiro sequences.

General framework applicable to polynomial and convex sequences.

Abstract

We consider the large sieve inequality for sparse sequences of moduli and give a general result depending on the additive energy (both symmetric and asymmetric) of the sequence of moduli. For example, in the case of monomials $f(X) = X^k$ this allows us to improve, in some ranges of the parameters, the previous bounds of S. Baier and L. Zhao (2005), K.~Halupczok (2012, 2015, 2018) and M.~Munsch (2020). We also consider moduli defined by polynomials $f(X) \in \mathbb{Z}[X]$, Piatetski-Shapiro sequences and general convex sequences. We then apply our results to obtain a version of the Bombieri--Vinogradov theorem with Piatetski-Shapiro moduli improving the level of distribution of R.~C.~Baker (2014).