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

For two distinct integers $m_1,m_2\ge 2$, 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 $$\{0\le n<N;S_{{\alpha }_1}(n)\equiv a_1(\mathrm{m}\mathrm{o}\mathrm{d}{b}_1),S_{{\alpha }_2}(n)\equiv a_2(\mathrm{m}\mathrm{o}\mathrm{d}{b}_2)\},$$ for all integers $a_1$ and $a_2\mathrm{.}$ 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).