A note on bounded exponential sums

TL;DR

This paper investigates the boundedness of exponential sums over subsets of natural numbers, proving such boundedness cannot hold for infinite non-cofinite sets across all real numbers, but can occur on large measure sets.

Contribution

It proves the non-existence of infinite non-cofinite sets with universally bounded exponential sums and characterizes sets where boundedness occurs on small intervals and rational points.

Findings

No infinite non-cofinite set has bounded exponential sums for all b1 in (0,1).

Sets with bounded sums on small intervals and rationals must be finite or cofinite.

Existence of infinite non-cofinite sets with bounded sums on large measure sets.

Abstract

Let $A\subset\mathbb{N}$, $\alpha\in(0,1)$, and for $x\in\mathbb{R}$ let $e(x):=e^{2\pi ix}$. We set $$S_{A}(\alpha,N):=\sum_{\substack{n\in A\n\leq N}}e(n\alpha).$$ Recently, Lambert A'Campo proposed the following question: is there an infinite non-cofinite set $A\subset\mathbb{N}$ such that for all $\alpha\in(0,1)$ the sum $S_{A}(\alpha,N)$ has bounded modulus as $N\to +\infty$? In this note we show that such sets do not exist. To do so, we use a theorem by Duffin and Schaeffer on complex power series. We extend our result by proving that if the sum $S_{A}(\alpha,N)$ is bounded in modulus on an arbitrarily small interval and on the set of rational points, then the set $A$ has to be either finite or cofinite. On the other hand, we show that there are infinite non-cofinite sets $A$ such that $|S_{A}(\alpha,N)|$ is bounded for all $\alpha\in E\subset (0,1)$, where $E$ has full Hausdorff dimension and $\mathbb{Q}\cap (0,1)\subset E$.