Erdens-Surányi序列与三角积分
Erdős-Surányi sequences and trigonometric integrals
摘要：
我们研究整数表示为$\pm a_1\pm a_2\pm \dotsb \pm a_n$形式的和，其中$a_1,a_2,\ldots$是一个规定的整数序列。这样一个序列被称为Erdens-Surányi序列，如果每个整数都可以写在这个形式的一些$n\in\mathbb{N}$和选择的迹象，在无限多的方式。我们研究了一个固定整数的表示数，它可以被写为三角积分，并获得了一个渐近公式下的一个相当一般的计划，由于罗斯和Szekeres。我们的方法，这是基于拉普拉斯近似积分的方法，也可以很容易地扩展到找到高阶扩展。作为推论，我们解决了Andrica和Ionaescu关于正负号方程$\pm 1^k \pm 2^k \pm \dotsb \pm n^k = 0$解的个数的一个猜想.
英文摘要：
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 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 $\pm 1^k \pm 2^k \pm \dotsb \pm n^k = 0$.
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