关于S'{a}rk"{o}zy和S'{o}s问题的一个注记

令$k,ell geq 2$是正整数.令$A$是无限非负整数的集合.对$nin mathbb{N}$, 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$ngeq 0$, $r_{1,k,ldots,k^{ell-1}}(A, n)=1$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.

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

Let $k,ell geq 2$ be positive integers. Let $A$ be an infinite set of nonnegative integers. For $nin mathbb{N}$, 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 $ngeq 0$ 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.