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

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).