On a conjecture of Dekking : The sum of digits of even numbers

Overview & Synopsis

Let q ≥ 2 and denote by sq the sum-of-digits function in base q. For j = 0, 1, . . . , q − 1 consider #{0 ≤ n < N : sq(2n) ≡ j (mod q)}. In 1983, F. M. Dekking conjectured that this quantity is greater than N/q and, respectively, less than N/q for infinitely many N , thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.