Fewest repetitions in infinite binary words

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 pmatch. 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 infi...

[hal-00742086, v1] 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 pmatch. 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 infi...

Binary words with the fewest unbordered conjugates

Avoiding large squares in infinite binary words

We consider three aspects of avoiding large squares in infinite binary words. First, we construct an infinite binary word avoiding both cubes xxx and squares yy with |y| ≥ 4; our construction is somewhat simpler than the original construction of Dekking. Second, we construct an infinite binary word avoiding all squares except 0, 1, and (01); our construction is somewhat simpler than the origina...

Extremal Infinite Overlap-Free Binary Words

Let t be the infinite fixed point, starting with 1, of the morphism μ : 0 → 01, 1→ 10. An infinite word over {0, 1} is said to be overlap-free if it contains no factor of the form axaxa, where a ∈ {0, 1} and x ∈ {0, 1}∗. We prove that the lexicographically least infinite overlap-free binary word beginning with any specified prefix, if it exists, has a suffix which is a suffix of t. In particula...