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

Abstract

Abstract 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 m 1,m 2 ≥ 2 be integers with gcd ( q - 1 , m 1 ) = gcd ( m , m 2 ) = 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 r 1,r 2, | { 0 ≤ n < N : S q ( n ) ≡ r 1 ( mod   m 1 ) ,     S α ( n ) ≡ r 2 ( mod   m 2 ) } | = N m 1 m 2 + 0 ( N 1 - 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.