A note on a problem of S\''{a}rk\"{o}zy and S\''{o}s 
Received:February 20, 2021 Revised:February 20, 2021 
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representation function linear form

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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
$r_{1,k,\ldots,k^{\ell1}} (A, n)$ denote the number of solutions of $n=a_0+ka_1+\cdots +k^{\ell1}a_{\ell1}$, $a_0, \ldots, a_{\ell1}\in A$.
In this paper, we show that $r_{1,k,\ldots,k^{\ell1}} (A, n)=1$ 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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