On digit sums of multiples of an integer
Let \(g\geq 2\) be an integer and \(s_g\) be the sum of digits in base \(g\). In the present paper the authors consider positive integers \(n\) such that \(s_g(n)\) and \(s_g(kn)\) satisfy certain relations for a fixed, or arbitrary positive integer \(k\). In their first result they study the set \[ \mathcal{N}_g=\left\{n:s_g(n)=s_g(kn)\text{ for some } k\neq g^{\ell}\right\}. \] In particular, they can show that the set \(\mathcal{N}_g\) contains all positive integers except for the powers of \(g\). The second result is an answer to a question raised by \textit{W. M. Schmidt} [Studies in pure mathematics, Mem. of P. Turán, 605--622 (1983; Zbl 0523.10030)], concerning for given \(K>0\) the numbers \(n\) such that \(s_2(n)\leq Ks_2(kn)\) for all \(k\geq1\). In particular, for \(x\geq 3\) they show that \[ \#\{n\leq x:s_g(n)\leq Ks_g(kn)\text{ for all }k\geq 1\}\ll\frac{x}{(\log x)^\frac12}. \]
scientific article; zbMATH DE number 5604006
scientific article