Subconvex $L^p$-sets, Weyl's inequality, and equidistribution

TL;DR

This paper studies special sets of natural numbers with subconvex bounds on exponential sums, establishing Weyl-type inequalities and exploring their implications for equidistribution, character sums, and arithmetic function averages.

Contribution

It introduces subconvex $L^p$-sets and proves Weyl-like inequalities for exponential sums over these sets, extending classical bounds to new contexts.

Findings

Weyl's inequality analogues for subconvex $L^p$-sets

Results on equidistribution of polynomial sequences in these sets

Applications to character sums and arithmetic function averages

Abstract

We examine sets $\mathscr A$ of natural numbers having the property that for some real number $p\in (0,2)$, one has the subconvex bound $$\int_0^1 \Bigl| \sum_{n\in \mathscr A\cap [1,N]}e(n\alpha)\Bigr|^p\, {\rm d}\alpha \ll N^{-1}|\mathscr A\cap [1,N]|^p.$$ We show that exponential sums over such sets satisfy inequalities analogous to Weyl's inequality, and in many circumstances of the same strength as classical versions of Weyl's bound. We also examine equidistribution of polynomials modulo $1$ in which the summands are restricted to these subconvex $L^p$-sets. In addition, we describe applications to problems involving character sums and averages of arithmetic functions.