Related papers: A note on univoque self-Sturmian numbers

Let $\beta \equiv \{ \beta_\mathbf{i} \}_{\mathbf{i} \in \mathbb{Z}_+^d}$ be a $d$-dimensional multisequence. Curto and Fialkow, have shown that if the infinite moment matrix $M(\beta)$ is finite-rank positive semidefinite, then $\beta$ has…

We study morphisms from certain classes and their action on episturmian words. The first class is $P_{ret}$. In general, a morphism of class $P_{ret}$ can map an infinite word having zero palindromic defect to a word having infinite…

Suppose $A\subset \mathbb{R}$ of size $k$ has distinct consecutive $r$--differences, that is for $1 \leq i \leq k -r$, the $r$--tuples $$(a_{i+1} - a_i , \ldots , a_{i+r} - a_{i + r -1})$$ are distinct. Then for any finite $B \subset…

When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$ respectively, with $|A| \geq \alpha X$ and $|B| \geq \beta X$, we show that the number of rational numbers expressible as $a/b$ with $(a,b)$ in $A \times B$ is $\gg (\alpha…

Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denotes the distance from $x$ to the nearest integer. In this paper we show that whenever $\alpha$ is irrational and $\beta$ is real then for any fixed $1<c<12/11$ there exist infinitely…

Unique expansions in non-integer bases $q$ have been investigated in many papers during the last thirty years. They are often conveniently generated by labeled directed graphs. In the first part of this paper we give a precise description…

Suppose that $\alpha_1, \alpha_2,\beta_1, \beta_2 \in\mathbb{R}$. Let $\alpha_1, \alpha_2 > 1$ be irrational and of finite type such that $1, \alpha_1^{-1}, \alpha_2^{-1}$ are linearly independent over $\mathbb{Q}$. Let $c$ be a real number…

Given two finite sequences of positive integers $\alpha$ and $\beta$, we associate a square free monomial ideal $I_{\alpha,\beta}$ in a ring of polynomials $S$, and we recursively compute the algebraic invariants of $S/I_{\alpha,\beta}$.…

A cryptarithm (or alphametic) is a mathematical puzzle in which numbers are represented with words in such a way that identical letters stand for equal digits and distinct letters for unequal digits. An alphametic puzzle is usually given in…

Consider $m \in \mathbb{N}$ and $\beta \in (1, m + 1]$. Assume that $a\in \mathbb{R}$ can be represented in base $\beta$ using a development in series $a = \sum^{\infty}_{n = 1}x(n)\beta^{-n}$ where the sequence $x = (x(n))_{n \in…

We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{\sigma(1)} \cdots x_{\sigma(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $\sigma$…

The Bernoulli convolution associated to the real $\beta>1$ and the probability vector $(p_0,..,p_{d-1})$ is a probability measure $\eta_{\beta,p}$ on $\mathbb R$, solution of the self-similarity relation…

In [Bak] the first author proved that for any $\beta\in (1,\beta_{KL})$ every $x\in(0,\frac{1}{\beta-1})$ has a simply normal $\beta$-expansion, where $\beta_{KL}\approx 1.78723$ is the Komornik-Loreti constant. This result is complemented…

We take a global view at substitution invariant Sturmian sequences. We show that homogeneous substitution invariant Sturmian sequences $s_{\alpha,\alpha}$ can be indexed by two binary trees, associated directly to Johannes Kepler's tree of…

In the several contexts such as combinatorial number theory, families of sets of positive integers closed under taking subsets have been investigated. Then it is sometimes useful to give bijections between the set of the one-sided infinite…

To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter…

Affine $\lambda$-terms are $\lambda$-terms in which each bound variable occurs at most once and linear $\lambda$-terms are $\lambda$-terms in which each bound variables occurs once. and only once. In this paper we count the number of closed…

An infinite binary sequence has randomness rate at least $\sigma$ if, for almost every $n$, the Kolmogorov complexity of its prefix of length $n$ is at least $\sigma n$. It is known that for every rational $\sigma \in (0,1)$, on one hand,…

We estimate multiplicative character sums taken on the values of a non-homogeneous Beatty sequence $\{[\alpha n + \beta] : n =1,2,... \}$, where $\alpha,\beta\in\R$, and $\alpha$ is irrational. Our bounds are nontrivial over the same short…

Let us call a simple graph on $n\geq 2$ vertices a prime gap graph if its vertex degrees are $1$ and the first $n-1$ prime gaps. We show that such a graph exists for every large $n$, and in fact for every $n\geq 2$ if we assume the Riemann…