'Research Institute for Mathematical Sciences, Kyoto University'
Abstract
This work proposes a new approach towards solving an over 20 years old conjecture regarding the maximum number of distinct squares that a word can contain. To this end we look at clusters of repetition roots, that is, the set of positions where the root u of a repetition u^[l] occurs. We lay the foundation of this theory by proving basic properties of these clusters and establishing upper bounds on the number of distinct squares when their roots form a chain with respect to the prefix order
