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

Abstract

Given two words u and υ, the binomial coefficient ( u υ ) is the number of ways υ appears as a subword (or subsequence) of u. The Thue-Morse sequence is the infinite word t= abbabaab⋯obtained by iteration of the morphism ɽ(a)=ab and ɽ(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 or equal to n, the binomial coefficient ( ɽ⌊ m ⌋ υ) is congruent to 0 mod p. In fact f( p, n)=2 n p 1+⌊ log p n⌋ for p≠2 and f(2, n)=2 k if F k −1 ⩽ n ⩽ F k , where F k denotes the kth 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.