Joint Distribution in Residue Classes of the Base-q and Ostrowski Digital Sums

Let q be an integer greater than or equal to 2, and let S_q(n)denote the sum of digits of n in base q.For

α=[0;1,m¯], m≥2,\alpha = \left[ {0;\overline {1,m} } \right],\,\,\,m \ge 2,

let S_α(n) denote the sum of digits in the Ostrowski α-representation of n. Let m₁,m₂ ≥ 2 be integers with

gcd(q-1,m1)=gcd(m,m2)=1\gcd \left( {q - 1,{m_1}} \right) = \gcd \left( {m,{m_2}} \right) = 1

We prove that there exists δ> 0 such that for all integers r₁,r₂,

|{0≤n<N:Sq(n)≡r1(mod m1), Sα(n)≡r2(mod m2)}|=Nm1m2+0(N1-δ).\matrix{ {\left| {\left\{ {0 \le n < N:{S_q}(n) \equiv {r_1}\left( {\bmod \,{m_1}} \right),\,\,{S_\alpha }(n) \equiv {r_2}\left( {\bmod \,{m_2}} \right)} \right\}} \right|} \cr { = {N \over {{m_1}{m_2}}} + 0\left( {{N^{1 - \delta }}} \right).} \cr }

The asymptotic relation implied by this equality was proved by Coquet, Rhin & Toffin and the equality was proved for the case α=[1¯]\alpha = \left[ {\bar 1} \right] by Spiegelhofer.