Density of k-ary words with 0, 1, 2 - error overlaps

Abstract

An overlap, or border, of a word is a prefix that is equal to the suffix of the same length. An overlap with q errors is a prefix which has distance q from the suffix of the same length; here, 0-error overlaps are classic ones. Unbordered, or bifix-free, words are a central notion in combinatorics on words and have a prominent role in many related areas, such as pattern matching or frame synchronization. On the other hand, words with 2-error overlaps arose as a characterization of isometric words, a notion recently introduced in the framework of hypercubes and their isometric subgraphs. This paper investigates the density of words with 0, 1, 2-error overlaps, where the words are taken over a generic k-ary alphabet, $k\ge 2$, and the distance they refer to is the Hamming or the Lee distance. Estimates on the limit density values are provided and compared in the case of binary and quaternary alphabets.