## Repetitions in words

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2024

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Repetitions are fundamental properties of words, and different types of repetitions have been explored in the area of word combinatorics. This thesis investigates two types of repetitions: squares and antisquares. We investigate the square conjecture that anticipates the number of distinct squares in a word is less than its length. It is known that any location of a word can be mapped to at most two rightmost squares, and a pair of these squares was referred to as an FS-double square. For simplicity, we will refer to the longer square in this pair as an FS-double square throughout this thesis. We examine the properties of words containing FS-double squares and explore the consecutive locations starting with FS-double squares. We observe that FS-double squares introduce no-gain locations where no rightmost squares ocCur. The count of these no-gain locations in words with a sequence of FS-double squares demonstrates that the square density of such words is less than 11/6. Furthermore, we investigate words that possess FS-double squares and maintain an equivalent number of such squares when reversed. We prove that the maximum number of FS-double squares in such a word is less than 1/11 th of the length of the word. Another aspect of our research involves counting squares in a repetition. A non-primitive word has a tom u for some non-empty word u and some positive integer k such that ko2. With no-gain locations and FS-double squares in these words, we conclude that the square density of such words approaches 1/2 as k increases. Also, we work on the lower bound of the square conjecture. The previous lower bound is obtained using a structure that generates words containing a high number of
distinct squares. We identify simiiar structures but produce words with more distinct squares. We also study antisquare, a special repetition of the form uiw here u is a binary word, and üis its complement. We show that a Word w can contain at most w((w|+2)/8 antisquares, and the lower bound for the number of distinct antisauares in w is lw-1.

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Supervisors: Kapoor, Kalpesh and Kenkireth, Benny George