A note on a problem of S\''{a}rk\"{o}zy and S\''{o}s
Received:February 20, 2021  Revised:February 20, 2021
Key Word: representation function   linear form  
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Min Tang School of Mathematics and Statistics, Anhui Normal University School of Mathematics and Computer Science,Anhui Normal University
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Abstract:
      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.
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