Related papers: Substitutions and Cantor real numeration systems
Given a sequence converging to zero, we consider the set of numbers which are sums of (infinite, finite, or empty) subsequences. When the original sequence is not absolutely summable, the subsum set is an unbounded closed interval which…
We associate in a canonical way a substitution to any abstract numeration system built on a regular language. In relationship with the growth order of the letters, we define the notion of two independent substitutions. Our main result is…
A numerical set $S$ is a cofinite subset of $\mathbb{N}$ which contains $0$. We use the natural bijection between numerical sets and Young diagrams to define a numerical set $\widetilde{S}$, such that their Young diagrams are complements.…
Dimension groups are complete invariants of strong orbit equivalence for minimal Cantor systems. This paper studies a natural family of minimal Cantor systems having a finitely generated dimension group, namely the primitive unimodular…
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…
We describe an algorithm for proving termination of programs abstracted to systems of monotonicity constraints in the integer domain. Monotonicity constraints are a non-trivial extension of the well-known size-change termination method.…
Based on the notion of thin sets introduced recently by T.~Banakh, Sz.~G\l{}\k{a}b, E.~Jab\l{}o\'nska and J.~Swaczyna we deliver a study of the infinite single-message transmission protocols. Such protocols are associated with a set of…
It is argued that transformation processes (generation rules) showing evidence of a long evolutionary history in universal computing systems can be generalized. The explicit function class $ \Omega $ is defined as follows: "Operators whose…
We provide a new way to represent numerical semigroups by showing that the position of every Ap\'ery set of a numerical semigroup $S$ in the enumeration of the elements of $S$ is unique, and that $S$ can be re-constructed from this…
We study automatic sequences and automatic systems generated by general constant length (nonprimitive) substitutions. While an automatic system is typically uncountable, the set of automatic sequences is countable, implying that most…
Given $\beta\in\mathbb{Z}[i]$ with $|\beta|>1$ and a finite set $D\subset\mathbb{Q}(i)$, let \[K_{\beta, D}=\left\{\sum_{j=1}^{\infty}\frac{d_j}{\beta^j}: d_j\in D, \forall j\geq 1\right\}.\] Let $\mathcal{S}$ be a finite set of…
For $\beta > 1$ a real algebraic integer ({\it the base}), the finite alphabets $\mathcal{A} \subset \mathbb{Z}$ which realize the identity $\mathbb{Q}(\beta) = {\rm Per}_{\mathcal{A}}(\beta)$, where ${\rm Per}_{\mathcal{A}}(\beta)$ is the…
We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle T. A set of integers is called r-Bohr if it is recurrent for all products of r rotations on T,…
An S-adic system is a symbolic dynamical system generated by iterating an infinite sequence of substitutions or morphisms, called a directive sequence. A finitary S-adic dynamical system is one where the directive sequence consists of…
Proving program termination is typically done by finding a well-founded ranking function for the program states. Existing termination provers typically find ranking functions using either linear algebra or templates. As such they are often…
We define Tate-Betti and Tate-Bass invariants for modules over a commutative noetherian local ring R. Then we show the periodicity of these invariants provided that R is a hypersurface. In case R is also Gorenstein, we show that a finitely…
Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by $T$, producing an asymptotic estimate as $T \to \infty$. This problem can be…
It has been known that there exists a canonical system for every finite real reflection group. The first and the third authors obtained an explicit formula for a canonical system in the previous paper. In this article, we first define…
We introduce a model-complete theory which completely axiomatizes the structure $Z_{\alpha}=(Z, +, 0, 1, f)$ where $f : x \to \lfloor{\alpha} x \rfloor $ is a unary function with $\alpha$ a fixed transcendental number. When $\alpha$ is…
We initiate the complexity theoretic study of the problem of computing the bits of (real) algebraic numbers. This extends the work of Yap on computing the bits of transcendental numbers like \pi, in Logspace. Our main result is that…