Distinct digits in base b expansions of linear recurrence sequences

Abstract

Let (u _n ) _n be a linear recurrence sequence of integers and let b > 1 be a natural number. In this paper, we show that under some mild technical assumptions the base b expansion of |u _n | has at least clog n/log log n non-zero digits when n is large, where c > 0 is a computable constant depending on the initial sequence (u _n ) _n and b. Our results complement the results of C.L. Stewart from [9]. Some diophantine applications are also presented.