Thue-Morse sequence and p-adic topology for the free monoid

Given two words u and v, the binomial coefficient ( u v ) is the number of ways v appears as a subword (or subsequence) of u. The Thue-Morse sequence is the infinite word t = abbabaab · · · obtained by iteration of the morphism τ (a) = ab and τ (b) = ba. We show that, for every prime p, and every positive integer n, there exists an integer m = f(p, n), such that, for every non-empty word v of length less than of equal to n, the binomial coefficient ( t[m] v ) is congruent to 0 modulo p. In fact f(p, n) = 2pp for p 6= 2 and f(2, n) = 2 if Fk−1 ≤ n < Fk, where Fk denotes the k-th Fibonacci number. It follows that, for each prime number p, there exists a sequence of left factors of t of increasing length, the limit of which is the empty word in the p-adic topology of the free monoid.