On exponential sums over multiplicative subgroups of medium size

TL;DR

This paper establishes new upper bounds for exponential sums over medium-sized multiplicative subgroups of finite fields and demonstrates their additive basis properties, also improving bounds for Heilbronn's sum.

Contribution

It introduces novel bounds for exponential sums over subgroups of sizes near the square root of the field size and applies these results to additive basis problems and Heilbronn's sum.

Findings

New upper bounds for exponential sums over subgroups

Subgroups of size > p^{1/2} form additive bases of order five

Improved bounds for Heilbronn's exponential sum

Abstract

In the paper we obtain some new upper bounds for exponential sums over multiplicative subgroups G of F^*_p having sizes in the range [p^{c_1}, p^{c_2}], where c_1,c_2 are some absolute constants close to 1/2. As an application we prove that in symmetric case G is always an additive basis of order five, provided by |G| > p^{1/2} log^{1/3} p. Also the method allows us to give a new upper bound for Heilbronn's exponential sum.