A repetition in a word is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in, that is those for which any extended subword of has a bigger period. The set of such repetitions represents in a compact way all repetitions in.We first prove a combinatorial result asserting that the sum of exponents of all maximal repetitions of a word of length is bounded by a linear function in. This implies, in particular, that there is only a linear number of maximal repetitions in a word. This allows us to construct a linear-time algorithm for finding all maximal repetitions. Some consequences and applications of these results are discussed, as well as related works. 1.

A (fractional) repetition in a word w is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in w, that is those for which any extended subword of w has a bigger period. The set of such repetitions represents in a compact way all repetitions in w. We first study maximal repetitions in Fibonacci words -- we count their exact number, and estimate the sum of their exponents. These quantities turn out to be linearly-bounded in the length of the word. We then prove that the maximal number of maximal repetitions in general words (on arbitrary alphabet) of length n is linearly-bounded in n, and we mention some applications and consequences of this result.

The problem of computing tandem repetitions with K possible mismatches is studied. Two main definitions are considered, and for both of them an O(nK log K + S) algorithm is proposed (S the size of the output). This improves, in particular, the bound obtained in [LS93]. Finally, other possible definions are briefly analyzed.

A string p = p0p1 · · · pn−1 of non-negative integers is a Euclidean string if the string (p0 + 1)p1 · · · (pn−1 − 1) is rotationally equivalent (i.e., conjugate) to p. We show that Euclidean strings exist if and only if n and p0 + p1 + · · · + pn−1 are relatively prime and that, if they exist, they are unique. We show how to construct them using an algorithm with the same structure as the Euclidean algorithm, hence the name. We show that Euclidean strings are Lyndon words and we describe relationships between Euclidean strings and the Stern-Brocot tree, Fibonacci strings, Beatty sequences, and Sturmian sequences. We also describe an application to a graph embedding problem.

We propose an algorithm for finding in a word all pairs of occurrences of the same subword with a given distance r between them. The obtained complexity is O(n log r + S), where S is the size of the output. We also show how the algorithm can be modified in order to find all such pairs of occurrences separated by a given word. The solution uses an algorithm for finding all quasi-squares in two strings, a problem that generalizes the known problem of searching for squares.