Squares and cubes in Sturmian sequences

Abstract

We prove that every Sturmian word ω has infinitely many prefixes of the form UnV 3 n , where |Un| < 2.855|Vn| and limn→∞ |Vn| = ∞. In passing, we give a very simple proof of the known fact that every Sturmian word begins in arbitrarily long squares.