The sum-of-digits function of polynomial sequences

Let q⩾2 be an integer and sq(n) denote the sum of the digits in base q of the positive integer n. The goal of this work is to study a problem of Gelfond concerning the re-partition of the sequence (sq(P(n)))n∈ℕ in arithmetic progressions when P∈ℤ[X] is such that P(ℕ)⊂ℕ. We answer Gelfond's question and we show the uniform distribution modulo 1 of the sequence (α sq(P(n)))n∈ℕ for α∈ℝ \ ℚ, provided that q is a large enough prime number co-prime with the leading coefficient of P.