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Sur la répartition jointe de la représentation d'Ostrowski dans les classes de résidue

For two distinct integers $m_1,m_2\ge2$, we set $α_1=[0;\overline{1,m_1}]$ and $α_2=[0;\overline{1,m_2}]$ and we denote by $S_{α_1}(n)$ and $S_{α_2}(n)$ respectively the sum of digits functions in the Ostrowski $α_1$ and $α_2-$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 $O(N^{1-δ})$ for the cardinal of the following set $$\Big\{ 0\leq n<N;\ S_{α_1}(n)\equiv a_1\pmod{b_1},\ S_{α_2}(n)\equiv a_2\pmod{b_2}\Big\},$$ for all integers $a_1$ and $a_2.$ Our result should be compared to that of Bésineau and Kim who treated the case of the $q-$representations in different bases (that are coprimes).