Sur la r\'{e}partition jointe de la repr\'{e}sentation d'Ostrowski dans les classes de r\'{e}sidue

TL;DR

This paper studies the joint distribution of Ostrowski digit sums for two specific irrational representations, providing an estimation with an error term for the count of integers with prescribed residue conditions.

Contribution

It extends previous work by estimating the distribution of digit sums in Ostrowski representations with explicit error bounds, generalizing results known for base-q representations.

Findings

Derived an $O(N^{1- ext{delta}})$ error estimate for the joint residue distribution.

Compared Ostrowski representation results with known base-q representation cases.

Established conditions under which the distribution estimates hold.

Abstract

For two distinct integers $m_1,m_2\ge2$, we set $\alpha_1=[0;\overline{1,m_1}]$ and $\alpha_2=[0;\overline{1,m_2}]$ and we denote by $S_{\alpha_1}(n)$ and $S_{\alpha_2}(n)$ respectively the sum of digits functions in the Ostrowski $\alpha_1$ and $\alpha_2-$representations of $n$. Let $b_1,b_2 $ be positive integers satisfying $(b_1,m_1)=1$ and $(b_2,m_2)=1$, we obtain an estimation with an error term $O(N^{1-\delta})$ for the cardinal of the following set $$\Big\{ 0\leq n<N;\ S_{\alpha_1}(n)\equiv a_1\pmod{b_1},\ S_{\alpha_2}(n)\equiv a_2\pmod{b_2}\Big\},$$ for all integers $a_1$ and $a_2.$ Our result should be compared to that of B\'{e}sineau and Kim who treated the case of the $q-$representations in different bases (that are coprimes).