Related papers: The Critical Exponent is Computable for Automatic …
The critical exponent of an infinite word $\bf x$ is the supremum, over all finite nonempty factors $f$, of the exponent of $f$. In this note we show that for all integers $k\geq 2,$ there is a binary infinite $k$-automatic sequence with…
For certain generalized Thue-Morse words t, we compute the "critical exponent", i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.
In this paper, we study the critical exponent of infinite words $\ubeta$ coding $\beta$-integers for $\beta$ being a~non-simple Parry number. In other words, we investigate the maximal consecutive repetitions of factors that occur in the…
The abelian critical exponent of an infinite word $w$ is defined as the maximum ratio between the exponent and the period of an abelian power occurring in $w$. It was shown by Fici et al. that the set of finite abelian critical exponents of…
We study how much injective morphisms can increase the repetitiveness of a given word. This question has a few possible variations depending on the meaning of ``repetitiveness''. We concentrate on fractional exponents of finite words and…
For each $\alpha > 2$ there is a binary word with critical exponent $\alpha$.
We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or…
We define and investigate the property of being `exponent-critical' for a finite group. A finite group is said to be exponent-critical if its exponent is not the least common multiple of the exponents of its proper non-abelian subgroups. We…
The first two authors have shown [KK99,KK00] that the sum the exponent (and thus the number) of maximal repetitions of exponent at least 2 (also called runs) is linear in the length of the word. The exponent 2 in the definition of a run may…
Over an alphabet of size 3 we construct an infinite balanced word with critical exponent 2+sqrt(2)/2. Over an alphabet of size 4 we construct an infinite balanced word with critical exponent (5+sqrt(5))/4. Over larger alphabets, we give…
Let $E$ be an elliptic curve defined over a number field $K$. We say that a prime number $p$ is exceptional for $(E,K)$ if $E$ admits a $p$-isogeny defined over $K$. The so-called exceptional set of all such prime numbers is finite if and…
In the classic problem of sequence prediction, a predictor receives a sequence of values from an emitter and tries to guess the next value before it appears. The predictor masters the emitter if there is a point after which all of the…
For any integer $k>2$, the infinite $k$-bonacci word $W^{(k)}$, on the infinite alphabet is defined as the fixed point of the morphism $\varphi_k:\mathbb{N}\rightarrow \mathbb{N}^2 \cup \mathbb{N}$, where \begin{equation*} \varphi_k(ki+j) =…
Cantor's ordinal numbers, a powerful extension of the natural numbers, are a cornerstone of set theory. They can be used to reason about the termination of processes, prove the consistency of logical systems, and justify some of the core…
We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth…
For an infinite word $x$, Bugeaud and Kim introduced a quantity $\mathrm{rep}(x)$ called the exponent of repetition of $x$. We prove that $\mathrm{rep}(x) = \mathrm{rep}(y)$ holds for a Sturmian word $x$ and every suffix $y$ of $x$. Let $c$…
We exhibit the construction of a deterministic automaton that, given k > 0, recognizes the (regular) language of k-differentiable words. Our approach follows a scheme of Crochemore et al. based on minimal forbidden words. We extend this…
In the last years the attention towards topological dynamical properties of highly discontinuous maps has increased significantly. In [D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus Math\'ematique, 360(G4),…
Deterministic one-way time-bounded multi-counter automata are studied with respect to their ability to perform reversible computations, which means that the automata are also backward deterministic and, thus, are able to uniquely step the…
The critical exponent $E(\mathbf u)$ of an infinite sequence $\mathbf u$ over a finite alphabet expresses the maximal repetition of a factor in $\mathbf u$. By the famous Dejean's theorem, $E(\mathbf u) \geq 1+\frac1{d-1}$ for every $d$-ary…