A Reverse Order Property of Correlation Measures of the Sum-of-digits Function

Let sq be the sum-of-digits function in base q, q � 2. If t is a positive integer, we denote by tR the unique integer that is obtained from t by reversing the order of the digits of the proper representation of t in base q. In this work we prove that for all α ∈ R and all positive integers t the correlation measure γ(α, t) = lim x→∞ 1 x � n<x e2πiα(sq(n+t)−sq(n)) satisfies γ(α, t) = γ(α, tR). From this we deduce that for all integers d the sets {n ∈ N : sq(n + t) − sq(n) = d} and {n ∈ N : sq(n + tR) − sq(n) = d} have the same asymptotic density. The proof involves methods coming from the study of q-additive functions, linear algebra, and analytic number theory.