Avoiding squares over words with lists of size three amongst four symbols

Rosenfeld, Matthieu

doi:10.48550/arxiv.2104.09965

Cited by 3 publications

(4 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The idea to find the sequence of coefficients is identical to the one used in Section 5 of [5]. We use a computer to first compute the set Λ of minimal forbidden factors and we can then find a set of coefficients with the desired properties.…”

Section: Verifying Lemma 3 and Lemmamentioning

confidence: 99%

“…Our proof relies on the same idea as the technique used in [5] (in fact, we recommend reading Lemma 2 of [5] before any proof from the current article since it is a less technical proof using the same central idea). More precisely, it relies on a recent counting argument [6,10] and on some ideas introduced by Koplakov and improved by Shur [2,7].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Ann wins the nonrepetitive game over four letters and the erase-repetition game over six letters

Rosenfeld¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We consider two games between two players Ann and Ben who build a word together by adding alternatively a letter at the end of the shared word. In the nonrepetitive game, Ben wins the game if he can create a square of length at least 4, and Ann wins if she can build an arbitrarily long word before that. In the erase-repetition game, whenever a square occurs the second part of the square is erased and the goal of Ann is still to build an arbitrarily large word (Ben simply wants to limit the size of the word in this game).Grytczuk, Kozik, and Micek showed that Ann has a winning strategy for the nonrepetitive game if the alphabet is of size at least 6 and for the erase-repetition game is the alphabet is of size at least 8. In this article, we lower these bounds to respectively 4 and 6. The bound obtain by Grytczuk et al. relied on the so-called entropy compression and the previous bound by Pegden relied on some particular version of the Lovász Local Lemma. We recently introduced a counting argument that can be applied to the same set of problems as entropy compression or the Lovász Local Lemma and we use our method here.For these two games, we know that Ben has a winning strategy when the alphabet is of size at most 3, so our result for the nonrepetitive game is optimal, but we are not able to close the gap for the erase-repetition game. * Supported by the ANR project CoCoGro (ANR-16-CE40-0005).

show abstract

Section: Verifying Lemma 3 and Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Ann wins the nonrepetitive game over four letters and the erase-repetition game over six letters

Rosenfeld¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The argument used for our lower bound is a simple counting technique recently introduced in [13] and already used in a few different settings [5,11,14,17]. In the setting of combinatorics on words a similar technique was already known under the name power series method [2,3,9,12].…”

Section: Introductionmentioning

confidence: 99%

Finding lower bounds on the growth and entropy of subshifts over countable groups

Rosenfeld¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We give a lower bound on the growth of a subshift based on a simple condition on the set of forbidden patterns defining that subshift. Aubrun et Al. showed a similar result based on the Lovász Local Lemma for subshift over any countable group and Bernshteyn extended their approach to deduce, amongst other things, some lower bound on the exponential growth of the subshift. However, our result has a simpler proof, is easier to use for applications, and provides better bounds on the applications from their articles (although it is not clear that our result is stronger in general).In the particular case of subshift over Z a similar but weaker condition given by Miller was known to imply nonemptiness of the associated shift. Pavlov used the same approach to provide a condition that implied exponential growth. We provide a version of our result for this particular setting and it is provably strictly stronger than the result of Pavlov and the result of Miller (and, in practice, leads to considerable improvement in the applications).We also apply our two results to a few different problems including strongly aperiodic subshifts, nonrepetitive subshifts, and Kolmogorov complexity of subshifts.

show abstract

“…Currently the best general result confirms it for alphabets of size 4 (see [13,12,26] for three different proofs). Recently Rosenfeld [27] proved that it holds when the union of all alphabets is a 4-element set.…”

Section: Introductionmentioning

confidence: 99%

Extensions and reductions of square-free words

Dębski¹,

Grytczuk²,

Pawlik³

2022

Preprint

View full text Add to dashboard Cite

A word is square-free if it does not contain a nonempty word of the form XX as a factor. A famous 1906 result of Thue asserts that there exist arbitrarily long squarefree words over a 3-letter alphabet. We study square-free words with additional properties involving single-letter deletions and extensions of words.A square-free word is steady if it remains square-free after deletion of any single letter. We prove that there exist infinitely many steady words over a 4-letter alphabet. We also demonstrate that one may construct steady words of any length by picking letters from arbitrary alphabets of size 7 assigned to the positions of the constructed word. We conjecture that both bounds can be lowered to 4, which is best possible.In the opposite direction, we consider square-free words that remain square-free after insertion of a single (suitably chosen) letter at every possible position in the word. We call them bifurcate. We prove a somewhat surprising fact, that over a fixed alphabet with at least three letters, every steady word is bifurcate. We also consider families of bifurcate words possessing a natural tree structure. In particular, we prove that there exists an infinite tree of doubly infinite bifurcate words over alphabet of size 12.

show abstract

Avoiding squares over words with lists of size three amongst four symbols

Cited by 3 publications

References 14 publications

Ann wins the nonrepetitive game over four letters and the erase-repetition game over six letters

Ann wins the nonrepetitive game over four letters and the erase-repetition game over six letters

Finding lower bounds on the growth and entropy of subshifts over countable groups

Extensions and reductions of square-free words

Contact Info

Product

Resources

About