Joint distribution in the residual classes of the sum of the numbers for two Ostrowski representations

For two distinct integers m(1), m(2) >= 2, we set alpha 1 = [0; (1, m(1)) over bar] and alpha(2) = [0; (1, m(2)) over bar] (where [0; (1,m) over bar] is the continued fraction [0; 1,m, 1,m, 1,m, ...]) and we let S-alpha 1(n) and S-alpha 2(n) denote 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 N/b(1)b(2) with an error term O(N1-delta) for the cardinality of the following set {0 <= n < N; S-alpha 1(n) equivalent to a(1) (modb(1)),S-alpha 2(n) equivalent to a(2) (modb(2))}, for all integers a(1) and a(2). Our result should be compared to that of Besineau and Kim who addressed the case of the q-representations in different bases (that are coprime).