Somme des chiffres et changement de base

TL;DR

This paper proves that for multiplicatively independent bases, the ratio of digit sums can approach any positive real number, and it provides bounds for related subsequences.

Contribution

It extends previous work by showing the density of digit sum ratios in the positive reals for multiplicatively independent bases.

Findings

Any positive real number is a limit point of the digit sum ratio sequence.

Provides upper and lower bounds for subsequence counting functions.

Extends previous results on digit sum ratios in different bases.

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.