Minimal weight expansions in Pisot bases

For applications to cryptography, it is important to represent numbers with a small number of non-zero digits (Hamming weight) or with small absolute sum of digits. The problem of finding representations with minimal weight has been solved for integer bases, e.g. by the non-adjacent form in base~2. In this paper, we consider numeration systems with respect to real bases $\beta$ which are Pisot numbers and prove that the expansions with minimal absolute sum of digits are recognizable by finite automata. When $\beta$ is the Golden Ratio, the Tribonacci number or the smallest Pisot number, we determine expansions with minimal number of digits $\pm1$ and give explicitely the finite automata recognizing all these expansions. The average weight is lower than for the non-adjacent form

Redundancy of minimal weight expansions in Pisot bases

Motivated by multiplication algorithms based on redundant number representations, we study representations of an integer $n$ as a sum $n=\sum_k \epsilon_k U_k$, where the digits $\epsilon_k$ are taken from a finite alphabet $\Sigma$ and $(U_k)_k$ is a linear recurrent sequence of Pisot type with $U_0=1$. The most prominent example of a base sequence $(U_k)_k$ is the sequence of Fibonacci numbers. We prove that the representations of minimal weight $\sum_k|\epsilon_k|$ are recognised by a finite automaton and obtain an asymptotic formula for the average number of representations of minimal weight. Furthermore, we relate the maximal order of magnitude of the number of representations of a given integer to the joint spectral radius of a certain set of matrices

Beta-expansions, natural extensions and multiple tilings associated with Pisot units

From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different transformations generating expansions in base $\beta$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $\beta$-expansions. Remarkably, the symmetric $\beta$-transformation does not satisfy this condition when $\beta$ is the smallest Pisot number or the Tribonacci number. This means that the Pisot conjecture on tilings cannot be extended to the symmetric $\beta$-transformation. Closely related to these (multiple) tilings are natural extensions of the transformations, which have many nice properties: they are invariant under the Lebesgue measure; under certain conditions, they provide Markov partitions of the torus; they characterize the numbers with purely periodic expansion, and they allow determining any digit in an expansion without knowing the other digits

A lower bound for Garsia's entropy for certain Bernoulli convolutions

Let $\beta\in(1,2)$ be a Pisot number and let $H_\beta$ denote Garsia's entropy for the Bernoulli convolution associated with $\beta$. Garsia, in 1963 showed that $H_\beta<1$ for any Pisot $\beta$. For the Pisot numbers which satisfy $x^m=x^{m-1}+x^{m-2}+...+x+1$ (with $m\ge2$) Garsia's entropy has been evaluated with high precision by Alexander and Zagier and later improved by Grabner, Kirschenhofer and Tichy, and it proves to be close to 1. No other numerical values for $H_\beta$ are known. In the present paper we show that $H_\beta>0.81$ for all Pisot $\beta$, and improve this lower bound for certain ranges of $\beta$. Our method is computational in nature.Comment: 16 pages, 4 figure

Finiteness property in Cantor real numeration systems

For alternate Cantor real base numeration systems we generalize the result of Frougny and~Solomyak on~arithmetics on the set of numbers with finite expansion. We provide a class of alternate bases which satisfy the so-called finiteness property. The proof uses rewriting rules on the~language of~expansions in the corresponding numeration system. The proof is constructive and provides a~method for~performing addition of~expansions in Cantor real bases. We consider a numeration system which is a common generalization of the positional systems introduced by Cantor and R\'enyi. Number representations are obtained using a composition of $\beta_k$-transformations for a given sequence of real bases $B=(\beta_k)_{k\geq 1}$, $\beta_k>1$. We focus on~arithmetical properties of the set of numbers with finite $B$-expansion in case that $B$ is an alternate base, i.e.\ $B$ is a periodic sequence. We provide necessary conditions for the so-called finiteness property. We further show a~sufficient condition using rewriting rules on the~language of~representations. The proof is constructive and provides a~method for~performing addition of~expansions in alternate bases. Finally, we give a family of alternate bases that satisfy this sufficient condition. Our work generalizes the results of Frougny and Solomyak obtained for the case when the base $B$ is a constant sequence.Comment: 19 pages

Nested quasicrystalline discretisations of the line

One-dimensional cut-and-project point sets obtained from the square lattice in the plane are considered from a unifying point of view and in the perspective of aperiodic wavelet constructions. We successively examine their geometrical aspects, combinatorial properties from the point of view of the theory of languages, and self-similarity with algebraic scaling factor $\theta$. We explain the relation of the cut-and-project sets to non-standard numeration systems based on $\theta$. We finally examine the substitutivity, a weakened version of substitution invariance, which provides us with an algorithm for symbolic generation of cut-and-project sequences

Deformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers

A connection between real poles of the growth functions for Coxeter groups acting on hyperbolic space of dimensions three and greater and algebraic integers is investigated. In particular, a geometric convergence of fundamental domains for cocompact hyperbolic Coxeter groups with finite-volume limiting polyhedron provides a relation between Salem numbers and Pisot numbers. Several examples conclude this work.Comment: 26 pages, 16 figures, 4 data tables; minor corrections; European Journal of Combinatorics, 201

Purity results for some arithmetically defined measures

We study measures that are obtained as push-forwards of measures of maximal entropy on sofic shifts under digital maps $(x_k)_{k\in\mathbb{N}}\mapsto\sum_{k\in\mathbb{N}}x_k\beta^{-k}$, where $\beta>1$ is a Pisot number. We characterise the continuity of such measures in terms of the underlying automaton and show a purity result