Let (M, d(x, y)) be a metric space. Thus M is a nonempty set and d(x, y) is a real-valued function defined for x, y ∈ M such that d(x, y) ≥ 0 for all x, y ∈ M , d(x, y) = 0 if and only if x = y, d(x, y) = d(y, x) for all x, y ∈ M , and d(x, z) ≤ d(x, y) + d(y, z) (1) for all x, y, z ∈ M. If we have the stronger condition that d(x, z) ≤ max(d(x, y), d(y, z)) (2) for all x, y, z ∈ M , then we say...

Given a domain $\Omega \subseteq \mathbb{R}^2$, let $\mathcal{D}(\Omega,x,R)$ be the number of lattice points from $\mathbb{Z}^2$ in $R\Omega-x$, for $R \ge 1$ and $x\in \mathbb{T}^2$, minus area $R\Omega$: $$\mathcal{D}(\Omega,x,R) = \# \{ (j,k) \in \mathbb{Z}^2 :(j-x_1,k-x_2) R\Omega \} - R^2|\Omega|.$$ We call $\int_{\mathbb{T}^2}|\mathcal{D}(\Omega,x,R)|^pdx$ $p$-th moment discrepancy funct...

The number theoretic analogue of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g ≥ 2, the study of h-nets in the additive group of integers with respect to the generating set Ag = {0}∪{±g : i = 0, 1, 2, . . .} requires a knowledge of the word lengths of integers with respect to Ag. A g-adic representation...

In this brief survey we give an introduction to some aspects of “atoms” on metric spaces and their connection with linear operators. Let (M, d(x, y)) be a metric space. For the purposes of this article, the underlying set M will always be assumed to contain at least 2 distinct elements. As usual, the distance function d(x, y) should satisfy d(x, y) ≥ 0 for all x, y ∈ M , d(x, y) = 0 if and only...

Let $(p_t)_{t\geq 0}$ be a convolution semigroup of probability measures on $\mathbb R^d$ defined by $$\int _{\mathbb R^d} e^{i\langle \xi ,x\rangle }\, p_t(\mathrm d x)= e^{-t\psi (\xi )}, \quad \ \in \mathbb R^d, $$ and let $\Omega $ an open subs

We start with the definitions of metric spaces and normed spaces. Definition 1 Let X be a set, and let d : X ×X → R ∪ {0}. The pair (X, d) is a metric space if for all x, y, z ∈ X, 1. d(x, y) = d(y, x) 2. d(x, y) = 0 ⇔ x = y 3. d(x, y) + d(y, z) ≥ d(x, z) (triangle inequality) Definition 2 A normed space is R for some finite k together with an associated mapping a → ‖a‖ from R to R ∪ {0} such t...

Let B be the closed unit ball in R and S its boundary. We define a family of pseudo metrics on B. As an application, we prove that for any countable-to-one function f : S → [0, a], the set NMnf = {x ∈ S 2 | there exists y ∈ S2 such that f(x)−f(y) > ndE(x, y)} is uncountable for all n ∈ N, where dE is the Euclidean metric on R . 2000 Mathematics Subject Classification ; 57N05, 57M40 1 The family...

We investigate the generalized convergence and sums of series of the form P n≥0 anT P (x), where P ∈ R[x], an ∈ R, ∀n ≥ 0, and T : R[x] → R[x] is a linear operator that commutes with the differentiation d dx : R[x]→ R[x]. CONTENTS 1. The main result 1 2. Some applications 3 References 9 1. THE MAIN RESULT We consider series of the form ∑ n≥0 anT P (x), (†) where P ∈ R[x], and T : R[x]→ R[x] is ...