Clusters of repetition roots forming prefix chains

We investigate lower bounds on the size of clusters (sets of starting positions of occurrences) of common prefixes shared by repetition roots. Such lower bounds in terms of the constituent roots in the sets provide upper bounds on the number of distinct repetitions. In the case of distinct square roots which are totally ordered by the prefix relation it has been shown that there must be more occurrences of the common prefix than the number of roots. Here we develop the theory further by presenting the tools to extend the bounds to exponents higher than 2 and we show that they are optimal in the sense that any sequence of cluster sizes satisfying the lower bounds can be realized. We also take the next step towards the bounds on arbitrary (only partially prefix-ordered) sets of roots by proving a lower bound on unbordered prefixes shared by two overlapping prefix chains of roots.