Exponential Sums and Distinct Points on Arcs

TL;DR

This paper extends previous results on exponential sums and point distributions on the unit circle, particularly focusing on points that are separated by a minimum distance, enhancing understanding of harmonic analysis in this context.

Contribution

The paper generalizes Lev's result by incorporating the case where points on the circle are bounded away from each other, building on recent related work.

Findings

Extended Freiman's lemma to arbitrary points separated by a minimum distance.

Provided new bounds for exponential sums with well-separated points.

Enhanced understanding of the structure of point sets on the unit circle.

Abstract

Suppose that some harmonic analysis arguments have been invoked to show that the indicator function of a set of residue classes modulo some integer has a large Fourier coefficient. To get information about the structure of the set of residue classes, we then need a certain type of complementary result. A solution to this problem was given by Gregory Freiman in 1961, when he proved a lemma which relates the value of an exponential sum with the distribution of summands in semi-circles of the unit circle in the complex plane. Since then, Freiman's Lemma has been extended by several authors. Rather than residue classes, one has considered the situation for finitely many arbitrary points on the unit circle. So far, Lev is the only author who has taken into consideration that the summands may be bounded away from each other, as is the case with residue classes. In this paper we extend Lev's result by lifting a recent result of ours to the case of the points being bounded away from each other.