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.