Palindrome Prefixes and Episturmian Words

Let w be an infinite word on an alphabet A. We denote by (ni)i≥1 the increasing sequence (assumed to be infinite) of all lengths of palindrome prefixes of w. In this text, we give an explicit construction of all words w such that ni+1 ≤ 2ni + 1 for all i, and study these words. Special examples include characteristic Sturmian words, and more generally standard episturmian words. As an application, we study the values taken by the quantity lim sup ni+1/ni, and prove that it is minimal (among all non-periodic words) for the Fibonacci word.