Additive energy and a large sieve inequality for sparse sequences

2021/03/23 by Baker, Roger C., Munsch, Marc, Shparlinski, Igor E. · 1 citation
#11D45 #11L07 #11N13 Secondary 11L15 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 11B57

paper · doi:10.48550/arxiv.2103.12659

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) ∈ 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).

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• Mean value theorems with smooth numbers

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