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Michel Dekking

Publications and source records attributed to Michel Dekking.

At least 19 recordsLinked to original sources
arXiv2023

On Hofstadter's G-sequence

We characterize the entries of Hofstadter's G-sequence in terms of the lower and upper Wythoff sequences. This can be used to give a short and comprehensive proof of the equality of Hofstadter's G-sequence and the sequence of averages of the swapped Wythoff sequences. In a second part we give some results that hold when one replaces the golden mean by other quadratic algebraic numbers. In a third part we prove a close relationship between Hofstadter's G-sequence and a sequence studied by Avdivpahic and Zejnulahi.

math.CO↗
arXiv2023

The interior of randomly perturbed self-similar sets on the line

Can we find a self-similar set on the line with positive Lebesgue measure and empty interior? Currently, we do not have the answer for this question for deterministic self-similar sets. In this paper we answer this question negatively for random self-similar sets which are defined with the construction introduced in the paper Jordan, Pollicott and Simon (Commun. Math. Phys., 2007). For the same type of random self-similar sets we prove the Palis-Takens conjecture which asserts that at least typically the algebraic difference of dynamically defined Cantor sets is either large in the sense that it contains an interval or small in the sense that it is a set of zero Lebesgue measure.

math.PR↗
arXiv2023

Counting base phi representations

In a base phi representation a natural number is written as a sum of powers of the golden mean $φ$. There are many ways to do this. How many? Even if the number of powers of $φ$ is finite, then any number has infinitely many base phi representations. By not allowing an expansion to end with the digits 0,1,1, the number of expansions becomes finite, a solution proposed by Ron Knott. Our first result is a recursion to compute this number of expansions. This recursion is closely related to the recursion given by Neville Robbins to compute the number of Fibonacci representations of a number, also known as Fibonacci partitions. We propose another way to obtain finitely many expansions, which we call the natural base phi expansions. We prove that these are closely connected to the Fibonacci partitions.

math.NT↗
arXiv2023

Two-block substitutions and morphic words

We consider in general two-block substitutions and their fixed points. We prove that some of them have a simple structure: their fixed points are morphic sequences. Others are intrinsically more complex, such as the Kolakoski sequence. We prove this for the Thue-Morse sequence in base 3/2.

math.CO↗
arXiv2023

The Thue-Morse sequence in base 3/2

We discuss the base 3/2 representation of the natural numbers. We prove that the sum of digits function of the representation is a fixed point of a 2-block substitution on an infinite alphabet, and that this implies that sum of digits function modulo 2 of the representation is a fixed point $x_{3/2}$ of a 2-block substitution on $\{0,1\}$. We prove that $x_{3/2}$ is mirror invariant, and present a list of conjectured properties of $x_{3/2}$, which we think will be hard to prove. Finally, we make a comparison with a variant of the base 3/2 representation, and give a general result on $p$-$q$-block substitutions.

math.CO↗
arXiv2021

On the representation of the natural numbers by powers of the golden mean

In a base phi representation a natural number is written as a sum of powers of the golden mean $φ$. There are many ways to do this. Well known is the standard representation, introduced by George Bergman in 1957, where a unique representation is obtained by requiring that no consecutive powers $φ^n$ and $φ^{n+1}$ do occur in the representation. In this paper we introduce a new representation by allowing that the powers $φ^0$ and $φ^1$ may occur at the same time, but no other consecutive powers. We then argue that this representation is much closer to the classical representation of the natural numbers by powers of an integer than Bergman's standard representation.

math.NT↗
arXiv2020

The sum of digits functions of the Zeckendorf and the base phi expansions

We consider the sum of digits functions for both base phi, and for the Zeckendorf expansion of the natural numbers. For both sum of digits functions we present morphisms on infinite alphabets such that these functions viewed as infinite words are letter-to-letter projections of fixed points of these morphisms. We characterize the first differences of both functions a) with generalized Beatty sequences, or unions of generalized Beatty sequences, and b) with morphic sequences.

math.CO↗
arXiv2020

The structure of Zeckendorf expansions

In this paper we classify the Zeckendorf expansions according to their digit blocks. It turns out that if we consider these digit blocks as labels on the Fibonacci tree, then the numbers ending with a given digit block in their Zeckendorf expansion appear as compound Wythoff sequences in a natural way on this tree. Here the digit blocks consisting of only $0$'s are an exception. We also give a second description of these occurrence sequences as generalized Beatty sequences. Finally, we characterize the numbers with a fixed digit block occurring at an arbitrary fixed position in their Zeckendorf expansions, and determine their densities.

math.CO↗
arXiv2019

The sum of digits function of the base phi expansion of the natural numbers

In the base phi expansion any natural number is written uniquely as a sum of powers of the golden mean with digits 0 and 1, where one requires that the product of two consecutive digits is always 0. In this paper we show that the sum of digits function modulo 2 of these expansions is a morphic sequence. In particular we prove that --- like for the Thue-Morse sequence --- the frequency of 0's and 1's in this sequence is equal to 1/2.

math.NT↗
arXiv2019

Morphic words, Beatty sequences and integer images of the Fibonacci language

Morphic words are letter-to-letter images of fixed points $x$ of morphisms on finite alphabets. There are situations where these letter-to-letter maps do not occur naturally, but have to be replaced by a morphism. We call this a decoration of $x$. Theoretically, decorations of morphic words are again morphic words, but in several problems the idea of decorating the fixed point of a morphism is useful. We present two of such problems. The first considers the so called $AA$ sequences, where $α$ is a quadratic irrational, $A$ is the Beatty sequence defined by $A(n)=\lfloor αn\rfloor$, and $AA$ is the sequence $(A(A(n)))$. The second example considers homomorphic embeddings of the Fibonacci language into the integers, which turns out to lead to generalized Beatty sequences with terms of the form $V(n)=p\lfloor αn\rfloor+qn+r$, where $p,q$ and $r$ are integers.

math.CO↗
arXiv2019

Base phi representations and golden mean beta-expansions

In the base phi representation any natural number is written uniquely as a sum powers of the golden mean with digits 0 and 1, where one requires that the product of two consecutive digits is always 0. In this paper we give precise expressions for the those natural numbers for which the $k$th digit is 1, proving two conjectures for $k=0,1$. The expressions are all in terms of generalized Beatty sequences.

math.NT↗
arXiv2017

The Frobenius problem for homomorphic embeddings of languages into the integers

Let S be a map from a language L to the integers satisfying S(vw)=S(v)+S(w) for all words v,w from the language. The classical Frobenius problem asks whether the complement of S(L) in the natural numbers will be infinite or finite, and in the latter case the value of the largest element in this complement. This is also known as the 'coin'-problem, and L is the full language consisting of all words over a finite alphabet. We solve the Frobenius problem for the golden mean language, any Sturmian language and the Thue-Morse language. We also consider two-dimensional embeddings.

math.CO↗
arXiv2017

Substitution invariant Sturmian words and binary trees

We take a global view at substitution invariant Sturmian sequences. We show that homogeneous substitution invariant Sturmian sequences $s_{α,α}$ can be indexed by two binary trees, associated directly to Johannes Kepler's tree of harmonic fractions from 1619. We obtain similar results for the inhomogeneous sequences $s_{α,1-α}$ and $s_{α,0}$.

math.CO↗
arXiv2016

On the conjugacy class of the Fibonacci dynamical system

We characterize the symbolical dynamical systems which are topologically isomorphic to the Fibonacci dynmaical system. We prove that there are infinitely many injective primitive substitutions generating a dynamical system in the Fibonacci conjugacy class. In this class there are infinitely many dynamical systems not generated by a substitution. An example is the system generated by doubling the 0's in the infinite Fibonacci word.

math.DS↗
arXiv2013

Diagonals in 2D-tilings and coincidence densities of substitutions

We study the diagonals of two-dimensional tilings generated by direct product substitutions. The properties of these diagonals are primarily determined by the eigenvalues of the substitution matrix, but also the order of the letters in the substitution plays a role. We show that the diagonals may fail to be uniformly recurrent, and that the frequencies of letters on the diagonal may not exist. We also highlight the connection with the density of coincidences and overlap distributions.

math.DS↗
arXiv2012

Plumes in kinetic transport: how the simple random walk can be too simple

We consider a discrete time particle model for kinetic transport on the two dimensional integer lattice. The particle can move due to advection in the $x$-direction and due to dispersion. This happens when the particle is free, but it can also be adsorbed and then does not move. When the dispersion of the particle is modeled by simple random walk, strange phenomena occur. In the second half of the paper, we resolve these problems and give expressions for the shape of the plume consisting of many particles.

math.PR↗