Exponential sums with automatic sequences

TL;DR

This paper demonstrates that automatic sequences are asymptotically orthogonal to certain periodic exponential functions, enabling new bounds and applications in number theory, including Kloosterman sums and congruence equations.

Contribution

It introduces a general method to bound sums over automatic sequences using two-point correlation sums, extending their analysis to exponential sums and congruences.

Findings

Automatic sequences are orthogonal to specific exponential functions.

Effective bounds on sums over automatic sequences are established.

Applications include bounds on Kloosterman sums and congruence solubility.

Abstract

We show that automatic sequences are asymptotically orthogonal to periodic exponentials of type $e_q(f(n))$, where $f$ is a rational fraction, in the P\'olya-Vinogradov range. This applies to Kloosterman sums, and may be used to study solubility of congruence equations over automatic sequences. We obtain this as consequence of a general result, stating that sums over automatic sequences can be bounded effectively in terms of two-point correlation sums over intervals.