Erd\H{o}s-Sur\'anyi sequences and trigonometric integrals

TL;DR

This paper investigates the representation of integers as sums with signs from prescribed sequences, providing asymptotic formulas for the number of such representations using trigonometric integrals and Laplace's method.

Contribution

It introduces a general approach to analyze Erdős–Surányi sequences and settles a conjecture on solutions to the signum equation using asymptotic analysis.

Findings

Derived asymptotic formulas for the number of representations

Extended the method to higher-order expansions

Settled a conjecture on the signum equation solutions

Abstract

We study representations of integers as sums of the form $\pm a_1\pm a_2\pm \dotsb \pm a_n$, where $a_1,a_2,\ldots$ is a prescribed sequence of integers. Such a sequence is called an Erd\H{o}s-Sur\'anyi 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 Iona\c{s}cu on the number of solutions to the signum equation $\pm 1^k \pm 2^k \pm \dotsb \pm n^k = 0$.