Joint distribution in residue classes of the base-q and Ostrowski digital sums

Abstract

Let q be an integer ≥2 and let Sq(n) denote the sum of digits of n in base q. For α=[0;1,m¯¯¯¯¯¯¯¯¯], m≥2, let Sα(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. We prove that there exists δ>0 such that for all integers a1,a2, |{0≤n<N:Sq(n)≡a1(modm1), Sα(n)≡a2(modm2)}|=Nm1m2+O(N1−δ). The asymptotic relation implied by this equality was proved by Coquet, Rhin & Toffin and the equality was proved for the case α=[ 1¯¯¯ ] by Spiegelhofer.