On subwords in the base-q expansion of polynomial and exponential functions

Let w be any word over the alphabet {0,1,…, q-1}, and denote by h either a polynomial of degree d≥ 1 or h: n↦ m^n for a fixed m. Furthermore, denote by e_q(w;h(n)) the number of occurrences of w as a subword in the base-q expansion of h(n). We show that lim sup_n→∞e_q(w;h(n))/log n≥γ(w)/llog q, where l is the length of w and γ(w)≥ 1 is a constant depending on a property of circular shifts of w. This generalizes work by the second author as well as is related to a generalization of Lagarias of a problem of Erdős.