Clusters of repetition roots: single chains

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 uℓ 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.