Fewest repetitions in infinite binary words

A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is the quotient of its length over its smallest period. In this article we give a proof of the fact that there exists an infinite binary word which contains finitely many squares and simultaneously avoids words of exponent larger than 7/3. Our infinite word contains 12 squares, which is the smallest possible number of squares to get the property, and 2 factors of exponent 7/3. These are the only factors of exponent larger than 2. The value 7/3 introduces what we call the finite-repetition threshold of the binary alphabet. We conjecture it is 7/4 for the ternary alphabet, like its repetitive threshold

Finite-Repetition threshold for infinite ternary words

The exponent of a word is the ratio of its length over its smallest period. The repetitive threshold r(a) of an a-letter alphabet is the smallest rational number for which there exists an infinite word whose finite factors have exponent at most r(a). This notion was introduced in 1972 by Dejean who gave the exact values of r(a) for every alphabet size a as it has been eventually proved in 2009. The finite-repetition threshold for an a-letter alphabet refines the above notion. It is the smallest rational number FRt(a) for which there exists an infinite word whose finite factors have exponent at most FRt(a) and that contains a finite number of factors with exponent r(a). It is known from Shallit (2008) that FRt(2)=7/3. With each finite-repetition threshold is associated the smallest number of r(a)-exponent factors that can be found in the corresponding infinite word. It has been proved by Badkobeh and Crochemore (2010) that this number is 12 for infinite binary words whose maximal exponent is 7/3. We show that FRt(3)=r(3)=7/4 and that the bound is achieved with an infinite word containing only two 7/4-exponent words, the smallest number. Based on deep experiments we conjecture that FRt(4)=r(4)=7/5. The question remains open for alphabets with more than four letters. Keywords: combinatorics on words, repetition, repeat, word powers, word exponent, repetition threshold, pattern avoidability, word morphisms.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Infinite binary words containing repetitions of odd period

A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is its length divided by its smallest period. In this article, we give some new results on the trade-off between the number of squares and the number of cubes in infinite binary words whose square factors have odd periods

Characterization of some binary words with few squares

Thue proved that the factors occurring infinitely many times in square-free words over {0,1,2} avoiding the factors in {010,212} are the factors of the fixed point of the morphism 0 → 012, 1 → 02, 2 → 1. He similarly characterized square-free words avoiding {010,020} and {121,212} as the factors of two morphic words. In this paper, we exhibit smaller morphisms to define these two square-free morphic words and we give such characterizations for six types of binary words containing few distinct squares

The Simplest Binary Word with Only Three Squares

We re-examine previous constructions of infinite binary words containing few distinct squares with the goal of finding the "simplest", in a certain sense. We exhibit several new constructions. Rather than using tedious case-based arguments to prove that the constructions have the desired property, we rely instead on theorem-proving software for their correctness

Words with many palindrome pair factors

Motivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many n, every length-n factor is a product of two palindromes. We show that every Sturmian word has this property, but this does not characterize the class of Sturmian words. We also show that the Thue-Morse word does not have this property. We investigate infinite words with the maximal number of distinct palindrome pair factors and characterize the binary words that are not palindrome pairs but have the property that every proper factor is a palindrome pair."The first author is supported by an NSERC USRA, the second by an NSERC Discovery Grant."http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p2