Upper bounds for double exponential sums along a subsequence

TL;DR

This paper extends bounds on double exponential sums along subsequences for all irrational numbers, generalizing previous results that were limited to badly-approximable irrationals, and provides a simplified proof.

Contribution

It offers a unified proof for bounds on double exponential sums applicable to all irrational numbers, broadening the scope beyond badly-approximable cases.

Findings

Established bounds for all irrational lpha

Simplified proof of Sinai and Ulcigrai's theorem

Generalized previous results to wider class of irrationals

Abstract

We consider a class of double exponential sums studied in a paper of Sinai and Ulcigrai. They proved a linear bound for these sums along the sequence of denominators in the continued fraction expansion of $\alpha$, provided $\alpha$ is badly-approximable. We provide a proof of a result, which includes a simple proof of their theorem, and which applies for all irrational $\alpha$.