Erdős-Surányi sequences and trigonometric integrals

Abstract

We study representations of integers as sums of the form $\pm a_1\pm a_2\pm \cdots \pm a_n$, where $a_1,a_2,\dots$ is a prescribed sequence of integers. Such a sequence is called an Erdős-Surányi sequence if every integer can be written in this form for some $n\in \mathbb{N}$ and choices of signs in infinitely many ways. We study the number of representations of a fixed integer, which can be written as a trigonometric integral, and obtain an asymptotic formula under a rather general scheme due to Roth and Szekeres. Our approach, which is based on Laplace's method for approximating integrals, can also be easily extended to find higher-order expansions. As a corollary, we settle a conjecture of Andrica and Ionaşcu on the number of solutions to the signum equation $\pm 1^k\pm 2^k\pm \cdots \pm n^k=0$.