DIGITAL SUM PROBLEMS AND SUBSTITUTIONS ON A FINITE ALPHABET

Let (ui)i ≥ 1 be a fixed point for a substitution σ on a finite alphabet A and for a ∈ A, f(a) a real number. We establish an asymptotic formula for S(N) = Σn < NΣi ≤ nf(ui) in the case where the second largest eigenvalue of the substitution matrix equals one and under some additional hypothesis. More precisely S(N) = αN logθ N + NF(N) + o(N), where the real number α depending on σ and f is explicitly determined and θ > 1 is the largest eigenvalue of the substitution matrix; F is a continuous, nowhere differentiable (if α∈0), real function such that F(θx) = F(x) for all x>0. Using the same method we prove a similar formula for Σn < N s(n), s(n) the sum of digits function with respect to the system of numeration associated with σ. These formulae generalize some recent work concerning digital sum problems. © 1991.