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Matthieu Rosenfeld

Publications and source records attributed to Matthieu Rosenfeld.

At least 19 recordsLinked to original sources
arXiv2026

Pairs of square-free arithmetic progressions in infinite words

We study a question of Harju from 2019 regarding the existence of infinite ternary square-free words whose subsequences modulo $p$ and $q$ are also square-free for relatively prime integers $p$ and $q$. Among such pairs $(p, q)$ with $p, q \geq 3$, the only two pairs with this property known prior to this work were $(3, 11)$ and $(5, 6)$. We prove that there are finitely many pairs $(p, q)$ of relatively prime integers with $p, q \geq 3$ for which there is no infinite ternary square-free word whose subsequences modulo $p$ and $q$ are square-free. To prove our result, we combine different techniques, including the construction of words from multi-valued square-free morphisms and circular square-free morphisms. We also introduce the notion of square-free transducers, a generalization of square-free morphisms that may be of independent interest.

math.CO↗
arXiv2025

There exist infinite cube-free words over any sequence of binary alphabets

We prove that for any sequence of binary alphabets $\mathcal{A}_1,\mathcal{A}_2,\dots$, there exists a cube-free word $c_1c_2\dots$ so that $c_1\in\mathcal{A}_1,c_2\in\mathcal{A}_2,\dots$. In particular, for every $n$, there are at least $1.35^n$ cube-free words in $\mathcal{A}_1\times\mathcal{A}_2\times\dots\times \mathcal{A}_n$. We also prove that if the list of alphabets is computable then one of these words is computable and its $n$th letter can be computed in time polynomial in $n$.

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arXiv2025

The lonely runner conjecture holds for eight runners

We prove that the lonely runner conjecture holds for eight runners. Our proof relies on a computer verification and on recent results that allow bounding the size of a minimal counterexample. We note that our approach also applies to the known cases with 4, 5, 6, and 7 runners. We expect that minor improvements to our approach could be enough to solve the cases of 9 or 10 runners.

math.CO↗
arXiv2025

Words avoiding the morphic images of most of their factors

We say that a finite factor $f$ of a word $w$ is \emph{imaged} if there exists a non-erasing morphism $m$, distinct from the identity, such that $w$ contains $m(f)$. We show that every infinite word contains an imaged factor of length at least 6 and that 6 is best possible. We show that every infinite binary word contains at least 36 distinct imaged factors and that 36 is best possible.

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arXiv2025

Decomposing a factorial into large factors

Let $t(N)$ denote the largest number such that $N!$ can be expressed as the product of $N$ integers greater than or equal to $t(N)$. The bound $t(N)/N = 1/e-o(1)$ was apparently established in unpublished work of Erd\H{o}s, Selfridge, and Straus; but the proof is lost. Here we obtain the more precise asymptotic $$ \frac{t(N)}{N} = \frac{1}{e} - \frac{c_0}{\log N} + O\left( \frac{1}{\log^{1+c} N} \right)$$ for an explicit constant $c_0 = 0.30441901\dots$ and some absolute constant $c>0$, answering a question of Erd\H{o}s and Graham. For the upper bound, a further lower order term in the asymptotic expansion is also obtained. With numerical assistance, we obtain highly precise computations of $t(N)$ for wide ranges of $N$, establishing several explicit conjectures of Guy and Selfridge on this sequence. For instance, we show that $t(N) \geq N/3$ for $N \geq 43632$, with the threshold shown to be best possible.

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arXiv2025

Local obstructions in sequences revisited

In this article, we consider some simple combinatorial game and a winning strategy in this game. This game is then used to prove several known results about non-repetitive sequences and approximations with denominators from a lacunary sequence. In this way we simplify the proofs, improve the bounds and get for free the computable versions that required a separate treatment.

math.CO↗
arXiv2024

An explicit condition for boundedly supermultiplicative subshifts

We study some properties of the growth rate of $\mathcal{L}(\mathcal{A},\mathcal{F})$, that is, the language of words over the alphabet $\mathcal{A}$ avoiding the set of forbidden factors $\mathcal{F}$. We first provide a sufficient condition on $\mathcal{F}$ and $\mathcal{A}$ for the growth of $\mathcal{L}(\mathcal{A},\mathcal{F})$ to be boundedly supermultiplicative. That is, there exist constants $C>0$ and $\alpha\ge0$, such that for all $n$, the number of words of length $n$ in $\mathcal{L}(\mathcal{A},\mathcal{F})$ is between $\alpha^n$ and $C\alpha^n$. In some settings, our condition provides a way to compute $C$, which implies that $\alpha$, the growth rate of the language, is also computable whenever our condition holds. We also apply our technique to the specific setting of power-free words where the argument can be slightly refined to provide better bounds. Finally, we apply a similar idea to $\mathcal{F}$-free circular words and in particular we make progress toward a conjecture of Shur about the number of square-free circular words.

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arXiv2024

Upper bounds on the average edit distance between two random strings

We study the average edit distance between two random strings. More precisely, we adapt a technique introduced by Lueker in the context of the average longest common subsequence of two random strings to improve the known upper bound on the average edit distance. We improve all the known upper bounds for small alphabets. We also provide a new implementation of Lueker technique to improve the lower bound on the average length of the longest common subsequence of two random strings for all small alphabets of size other than $2$ and $4$.

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arXiv2024

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

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 $\mathbb{Z}$ 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.

math.DS↗
arXiv2023

On Vizing's problem for triangle-free graphs

We prove that $\chi(G) \le \lceil (\Delta+1)/2\rceil+1$ for any triangle-free graph $G$ of maximum degree $\Delta$ provided $\Delta \ge 524$. This gives tangible progress towards an old problem of Vizing, in a form cast by Reed. We use a method of Hurley and Pirot, which in turn relies on a new counting argument of the second author.

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arXiv2023

Reconstructing words using queries on subwords or factors

We study word reconstruction problems. Improving a previous result by P. Fleischmann, M. Lejeune, F. Manea, D. Nowotka and M. Rigo, we prove that, for any unknown word $w$ of length $n$ over an alphabet of cardinality $k$, $w$ can be reconstructed from the number of occurrences as subwords (or scattered factors) of $O(k^2\sqrt{n\log_2(n)})$ words. Two previous upper bounds obtained by S. S. Skiena and G. Sundaram are also slightly improved: one when considering information on the existence of subwords instead of on the numbers of their occurrences, and, the other when considering information on the existence of factors.

cs.DM↗
arXiv2022

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

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.

math.CO↗
arXiv2022

It is undecidable whether the growth rate of a given bilinear system is 1

We show that there exists no algorithm that decides for any bilinear system $(B,v)$ if the growth rate of $(B,v)$ is $1$. This answers a question of Bui who showed that if the coefficients are positive the growth rate is computable (i.e., there is an algorithm that outputs the sequence of digits of the growth rate of $(B,v)$). Our proof is based on a reduction of the computation of the joint spectral radius of a set of matrices to the computation of the growth rate of a bilinear system. We also use our reduction to deduce that there exists no algorithm that approximates the growth rate of a bilinear system with relative accuracy $\varepsilon$ in time polynomial in the size of the system and of $\varepsilon$. Our two results hold even if all the coefficients are nonnegative rationals.

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arXiv2021

Avoiding Square-Free Words on Free Groups

We consider sets of factors that can be avoided in square-free words on two-generator free groups. The elements of the group are presented in terms of 0,1,2,3 such that 0 and 2 (resp.,1 and 3) are inverses of each other so that 02, 20, 13 and 31 do not occur in a reduced word. A Dean word is a reduced word that does not contain occurrences of $uu$ for any nonempty $u$. Dean showed in 1965 that there exist infinite square-free reduced words. We show that if $w$ is a Dean word of length at least 59 then there are at most six reduced words of length 3 avoided by $w$. We construct an infinite Dean word avoiding six reduced words of length~3. We also construct infinite Dean words with low critical exponent and avoiding fewer reduced words of length 3. Finally, we show that the minimal frequency of a letter in a Dean word is $8/59$ and the growth rate is close to 1.45818.

math.CO↗
arXiv2021

Avoiding large squares in trees and planar graphs

The Thue number $π(G)$ of a graph $G$ is the minimum number of colors needed to color $G$ without creating a square on a path of $G$. For a graph class $C$, $π(C)$ is the supremum of $π(G)$ over the graphs $G\in C$. The Thue number has been investigated for famous minor-closed classes: $π(tree)=4$, $7\leπ(outerplanar)\le12$, and $11\leπ(planar)\le768$. Following a suggestion of Grytczuk, we consider the generalized parameters $π_k(C)$ such that only squares of period at least $k$ must be avoided. Thus, $π(C)=π_1(C)$. We show that $π_5(tree)=2$, $π_2(tree)=3$, and $π_k(planar)\ge11$ for every fixed $k$.

math.CO↗
arXiv2021

Lower-bounds on the growth of power-free languages over large alphabets

We study the growth rate of some power-free languages. For any integer $k$ and real $β>1$, we let $α(k,β)$ be the growth rate of the number of $β$-free words of a given length over the alphabet $\{1,2,\ldots, k\}$. Shur studied the asymptotic behavior of $α(k,β)$ for $β\ge2$ as $k$ goes to infinity. He suggested a conjecture regarding the asymptotic behavior of $α(k,β)$ as $k$ goes to infinity when $1<β<2$. He showed that for $\frac{9}{8}\leβ<2$ the asymptotic upper-bound holds of his conjecture holds. We show that the asymptotic lower-bound of his conjecture holds. This implies that the conjecture is true for $\frac{9}{8}\leβ<2$.

math.CO↗
arXiv2021

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

In 2007, Grytczuk conjecture that for any sequence $(\ell_i)_{i\ge1}$ of alphabets of size $3$ there exists a square-free infinite word $w$ such that for all $i$, the $i$-th letter of $w$ belongs to $\ell_i$. The result of Thue of 1906 implies that there is an infinite square-free word if all the $\ell_i$ are identical. On the other, hand Grytczuk, Przybyło and Zhu showed in 2011 that it also holds if the $\ell_i$ are of size $4$ instead of $3$. In this article, we first show that if the lists are of size $4$, the number of square-free words is at least $2.45^n$ (the previous similar bound was $2^n$). We then show our main result: we can construct such a square-free word if the lists are subsets of size $3$ of the same alphabet of size $4$. Our proof also implies that there are at least $1.25^n$ square-free words of length $n$ for any such list assignment. This proof relies on the existence of a set of coefficients verified with a computer. We suspect that the full conjecture could be resolved by this method with a much more powerful computer (but we might need to wait a few decades for such a computer to be available).

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