Self-generating sets, integers with missing blocks, and substitutions

AbstractWe give a new construction of the Kimberling sequence defined by:(a)1 belongs to S;(b)if the positive integer x belongs to S, then 2x and 4x-1 belong to S; and(c)nothing else belongs to S,hence, S=12346781112141516…which is sequence A052499 in the Sloane's On-line Encyclopedia of Integer Sequences, by proving that this sequence is equal to sequence 1+ A003754, the sequence of integers whose binary expansion does not contain the block of digits 00. We give a general framework for this sequence and similar sequences, in relation to automatic or morphic sequences and to non-standard numeration systems such as the lazy Fibonacci expansion

A Probabilistic Approach to Generalized Zeckendorf Decompositions

Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-$b$ expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogs of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts.Comment: Version 1.0, 25 pages. Keywords: Zeckendorf decompositions, positive linear recurrence relations, distribution of gaps, longest gap, Markov processe

The minimal automaton recognizing mN in a linear numeration system

We study the structure of automata accepting the greedy representations of N in a wide class of numeration systems. We describe the conditions under which such automata can have more than one strongly connected component and the form of any such additional components. Our characterization applies, in particular, to any automaton arising from a Bertrand numeration system. Furthermore, we show that for any automaton A arising from a system with a dominant root β > 1, there is a morphism mapping A onto the automaton arising from the Bertrand system associated with the number β. Under some mild assumptions, we also study the state complexity of the trim minimal automaton accepting the greedy representations of the multiples of m ≥ 2 for a wide class of linear numeration systems. As an example, the number of states of the trim minimal automaton accepting the greedy representations of mN in the Fibonacci system is exactly 2m2

Numeration Systems: a Link between Number Theory and Formal Language Theory

We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal interpretations and some research directions. We discuss the notion of numeration systems, recognizable sets of integers and automatic sequences. We briefly sketch some results about transcendence related to the representation of real numbers. We conclude with some applications to combinatorial game theory and verification of infinite-state systems and present a list of open problems.Comment: 21 pages, 3 figures, invited talk DLT'201

Automatic sequences based on Parry or Bertrand numeration systems

We study the factor complexity and closure properties of automatic sequences based on Parry or Bertrand numeration systems. These automatic sequences can be viewed as generalizations of the more typical $k$-automatic sequences and Pisot-automatic sequences. We show that, like $k$-automatic sequences, Parry-automatic sequences have sublinear factor complexity while there exist Bertrand-automatic sequences with superlinear factor complexity. We prove that the set of Parry-automatic sequences with respect to a fixed Parry numeration system is not closed under taking images by uniform substitutions or periodic deletion of letters. These closure properties hold for $k$-automatic sequences and Pisot-automatic sequences, so our result shows that these properties are lost when generalizing to Parry numeration systems and beyond. Moreover, we show that a multidimensional sequence is $U$-automatic with respect to a positional numeration system $U$ with regular language of numeration if and only if its $U$-kernel is finite.</p

Ultimate periodicity problem for linear numeration systems

We address the following decision problem. Given a numeration system U and a U-recognizable subset X of N, i.e. the set of its greedy U-representations is recognized by a finite automaton, decide whether or not X is ultimately periodic. We prove that this problem is decidable for a large class of numeration systems built on linear recurrence sequences. Based on arithmetical considerations about the recurrence equation and on p-adic methods, the DFA given as input provides a bound on the admissible periods to test