Random exponential sums and lattice points in regions

TL;DR

This paper investigates exponential sums with randomized frequencies, connecting problems in harmonic analysis, number theory, and lattice point counting, and offers new insights into classical problems like the Dirichlet divisor problem.

Contribution

It introduces a stochastic approach to analyze exponential sums, establishing new links between harmonic analysis and lattice point problems in number theory.

Findings

New bounds for exponential sums with randomized frequencies

Connections established between harmonic analysis and lattice point counting

Insights into classical problems like the Dirichlet divisor problem

Abstract

In this article we study two fundamental problems on exponential sums via randomization of frequencies with stochastic processes. These are the Hardy-Littlewood majorant problem, and $L^{2n}(\mathbb{T}), \ n\in \mathbb{N}$ norms of exponential sums, which can also be interpreted as solutions of diophantine equations or lattice points on surfaces. We establish connections to the well known problems on lattice points in regions such as the Dirichlet divisor problem.