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

https://doi.org/10.1016/0012-365X(89)90302-6Get rights and content
Under an Elsevier user license
open archive

Abstract

Given two words u and υ, the binomial coefficient (uυ) 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⌊mυ) is congruent to 0 mod p. In fact f(p, n)=2np1+⌊logpn for p≠2 and f(2, n)=2k if Fk−1nFk, where Fk 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.

Cited by (0)

Research on this paper was partially supported by PRC “Mathématique et Informatique”.