Joint Distribution in Residue Classes of the Base- q and Ostrowski Digital Sums
Divyum Sharma · Uniform distribution theory · 2019
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 (