Squares and cubes in Sturmian sequences

Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius 03225, Lithuania; arturas.dubickas@mif.vu.lt

Received: 1 December 2008
Accepted: 5 February 2009

Abstract

We prove that every Sturmian word ω has infinitely many prefixes of the form U_nV_n³, where |U_n| < 2.855|V_n| and lim_n→∞|V_n| = ∞. In passing, we give a very simple proof of the known fact that every Sturmian word begins in arbitrarily long squares.