The sum‐of‐digits function of polynomial sequences

Let q ⩾2 be an integer and s q ( 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 ( s q ( 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 s q ( 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 .