Erdos--Suranyi sequences and trigonometric integrals

Abstract

We study representations of integers as sums of the form ±a1 ± a2 ± · · · ± an, where a1, a2, . . . 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 ∈ 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 ±1 k ± 2 k ± · · · ± n k = 0.