Additive energy and a large sieve inequality for sparse 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 $f(X) = X^k$ 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 ) ∈ Z [ X ] $f(X) \in \mathbb {Z}[X]$ , 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).