A Note on a Problem of S\'{a}rk\"{o}zy and S\'{o}s

DOI：10.3770/j.issn:2095-2651.2022.03.003

 作者 单位 汤敏 安徽师范大学数学与统计学院, 安徽 芜湖 241002

令$k,\ell \geq 2$是正整数.令$A$是无限非负整数的集合.对$n\in \mathbb{N}$, 令$r_{1,k,\ldots,k^{\ell-1}}(A, n)$表示方程$n=a_0+ka_1+\cdots +k^{\ell-1}a_{\ell-1}$, $a_0, \ldots, a_{\ell-1}\in A$解的个数. 在本文中, 我们证明了对所有$n\geq 0$, $r_{1,k,\ldots,k^{\ell-1}}(A, n)=1$当且仅当$A$是$k^\ell$进制展开中数位小于$k$的所有非负整数的集合. 这个结果部分回答了S\'{a}rk\"{o}zy and S\'{o}s关于多维线性型表示的一个问题.

Let $k,\ell \geq 2$ be positive integers. Let $A$ be an infinite set of nonnegative integers. For $n\in \mathbb{N}$, let $r_{1,k,\ldots,k^{\ell-1}} (A, n)$ denote the number of solutions of $n=a_0+ka_1+\cdots +k^{\ell-1}a_{\ell-1}$, $a_0, \ldots, a_{\ell-1}\in A$. In this paper, we show that $r_{1,k,\ldots,k^{\ell-1}} (A, n)=1$ for all $n\geq 0$ if and only if $A$ is the set of all nonnegative integers such that all its digits in its $k^\ell$-adic expansion are smaller than $k$. This result partially answers a question of S\'{a}rk\"{o}zy and S\'{o}s on representation for multivariate linear forms.