A Note on a Problem of S\'{a}rk\"{o}zy and S\'{o}s
Received:February 20, 2021  Revised:May 20, 2021
Key Word: representation function   linear form
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11971033), Top Talents Project of Anhui Department of Education (Grant No.gxbjZD05) and the Natural Science Foundation of Anhui Province (Grant No.2008085QA06).
 Author Name Affiliation Min TANG School of Mathematics and Statistics, Anhui Normal University, Anhui 241002, P. R. China
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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.