Monochromatic arithmetic progressions in automatic sequences with group structure

We determine asymptotic growth rates for lengths of monochromatic arithmetic progressions in certain automatic sequences. In particular, we look at (one-sided) fixed points of aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue-Morse and Rudin-Shapiro substitutions, respectively. For such infinite words, we show that there exists a subsequence {dn} of differences along which the maximum length A (dn) of a monochromatic arithmetic progression (with fixed difference dn) grows at least polynomially in dn. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution. As an application, we obtain bounds for a van der Waerden-type number for a class of colourings parametrised by the size of the alphabet and the length of the substitution

In Memoriam, Solomon Marcus

This book commemorates Solomon Marcus’s fifth death anniversary with a selection of articles in mathematics, theoretical computer science, and physics written by authors who work in Marcus’s research fields, some of whom have been influenced by his results and/or have collaborated with him

Automatic Sequences and Decidable Properties: Implementation and Applications

In 1912 Axel Thue sparked the study of combinatorics on words when he showed that the Thue-Morse sequence contains no overlaps, that is, factors of the form ayaya. Since then many interesting properties of sequences began to be discovered and studied. In this thesis, we consider a class of infinite sequences generated by automata, called the k-automatic sequences. In particular, we present a logical theory in which many properties of k-automatic sequences can be expressed as predicates and we show that such predicates are decidable. Our main contribution is the implementation of a theorem prover capable of practically characterizing many commonly sought-after properties of k-automatic sequences. We showcase a panoply of results achieved using our method. We give new explicit descriptions of the recurrence and appearance functions of a list of well-known k-automatic sequences. We define a related function, called the condensation function, and give explicit descriptions for it as well. We re-affirm known results on the critical exponent of some sequences and determine it for others where it was previously unknown. On the more theoretical side, we show that the subword complexity p(n) of k-automatic sequences is k-synchronized, i.e., the language of pairs (n, p(n)) (expressed in base k) is accepted by an automaton. Furthermore, we prove that the Lyndon factorization of k-automatic sequences is also k-automatic and explicitly compute the factorization for several sequences. Finally, we show that while the number of unbordered factors of length n is not k-synchronized, it is k-regular

Forward Limit Sets of Semigroups of Substitutions and Arithmetic Progressions in Automatic Sequences

This thesis deals with symbolic sequences generated by semigroups of substitutions acting on finite alphabets. First, we investigate the underlying structure of certain automatic sequences by studying the maximum length A(d) of the monochromatic arithmetic progressions of difference d appearing in these sequences. For example, for the Thue-Morse sequence and a class of generalised Thue-Morse sequences, we give exact values of A(d) or upper bounds on it, for certain differences d. For aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue-Morse and Rudin-Shapiro substitutions, respectively, we study the asymptotic growth rate of A(d). In particular, we prove that there exists a subsequence (d_n) of differences along which A(d_n) grows at least polynomially in d_n. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution considered. Next, we introduce the forward limit set Λ of a semigroup S generated by a family of substitutions of a finite alphabet, which typically coincides with the set of all possible s-adic limits of that family. We provide several alternative characterisations of the forward limit set. For instance, we prove that Λ is the unique maximal closed and strongly S-invariant subset of the space of all infinite words, and we prove that it is the closure of the image under S of the set of all fixed points of S. It is usually difficult to compute a forward limit set explicitly; however, we show that, provided certain assumptions hold, Λ is uncountable, and we supply upper bounds on its size in terms of logarithmic Hausdorff dimension

On asymptotically automatic sequences

30 pages; comments welcomeWe study the notion of an asymptotically automatic sequence, which generalises the notion of an automatic sequence. While $k$-automatic sequences are characterised by finiteness of $k$-kernels, the $k$-kernels of asymptotically $k$-automatic sequences are only required to be finite up to equality almost everywhere. We prove basic closure properties and a linear bound on asymptotic subword complexity, show that results concerning frequencies of symbols are no longer true for the asymptotic analogue, and discuss some classification problems

On asymptotically automatic sequences

We study the notion of an asymptotically automatic sequence, which generalises the notion of an automatic sequence. While $k$-automatic sequences are characterised by finiteness of $k$-kernels, the $k$-kernels of asymptotically $k$-automatic sequences are only required to be finite up to equality almost everywhere. We prove basic closure properties and a linear bound on asymptotic subword complexity, show that results concerning frequencies of symbols are no longer true for the asymptotic analogue, and discuss some classification problems.Comment: 30 pages; early draft, comments welcom