On a family of sparse exponential sums

TL;DR

This paper studies exponential sums with sparse polynomial phases modulo primes, providing explicit bounds and exploring their value distribution, with applications to quantum chaos models like the quantum cat map.

Contribution

It offers explicit bounds for sparse exponential sums and initiates their value distribution analysis, extending Bourgain's earlier estimates.

Findings

Derived explicit bounds with savings over averages

Analyzed the value distribution of these sums

Connected results to quantum chaos models

Abstract

We investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear in the study of the quantum cat map. While they are not amenable to treatment by algebro-geometric methods such as Weil's bounds, Bourgain (2005) gave a nontrivial estimate for these and more general sums. In this work we obtain explicit bounds with reasonable savings over various types of averaging. We also initiate the study of the value distribution of these sums.