Somme des chiffres et changement de base

Abstract

Let $s_a(n)$ denote the sum of digits of an integer $n$ in the base $a$ expansion. Answering, in a extended form, a question of Deshouillers, Habsieger, Laishram, and Landreau, we show that, provided $a$ and $b$ are multiplicatively independent, any positive real number is a limit point of the sequence $\{s_b(n)/s_a(n)\}_{n=1}^{\infty}$. We also provide upper and lower bounds for the counting functions of the corresponding subsequences.