Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2511.18064

Densitometria I. Discrete groups

Szil\' ard Gy. R\' ev\' esz·Imre Z. Ruzsa
View original source ↗Read source document ↗Export metadata ↓

Abstract

An upper mean here is a subadditive functional $\overline M$ defined on bounded functions on a commutative group which has, beside some natural requirements, the property we call restricted additivity: if $g(x)= f(x)+f(x+t)$, then $\overline{M} (g)= 2 \overline{M} (f)$. This tries to grasp that it should not depend on local properties. This naturally induces a lower mean, and when they coincide it is the mean. Restriction to 0--1 valued functions (sets) is a density. We answer the following questions: Given a functional defined on a subset of all functions, when is it a mean? Given a functional, which is a mean, how do we find the upper mean it came from? Is it unique? Given a function $f$, what are the possible values of $\overline M(f)$, for upper means $\overline M$? In particular, we find the extremal means and give several expressions for it. We propose the names ``lowest and uppermost mean'' for them to replace the not really justified names ``lower and upper Banach mean and density''. We also consider analogous questions for densities, with partial answers only.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Szil\' ard Gy. R\' ev\' esz, Imre Z. Ruzsa. 2025-11-22. Densitometria I. Discrete groups. https://arxiv.org/abs/2511.18064

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Regular specular differentiation in Euclidean spaces

We study the regular specular derivative, a generalized derivative defined at every point where both one-sided derivatives exist and are finite. Geometrically, it is the slope of the mirror that reflects the left tangent ray into the right one. In one variable we derive computational formulas, prove inverse function and rotation rules, establish Quasi-Rolle's Theorem and the Quasi-Mean Value Theorem, and obtain a derivative-limit theorem, which shows that twice regularly specularly differentiable functions are continuously differentiable. We also prove both parts of the Fundamental Theorem of Calculus. In several variables we introduce specular gradients, directional derivatives, tangent hyperplanes, and normal vectors, show that a continuous specular gradient forces classical differentiability, and characterize when the specular tangent hyperplane is unique.

math.CA↗
arXiv2026

Prevalent smoothness in inhomogeneous Besov spaces

In this article, we prove that, under some assumptions on the so-called environment, prevalent functions in inhomogeneous Besov spaces recently introduced by Barral-Seuret in 2023 are multifractal, with a singularity spectrum that we determine. This completes the previous Baire generic results already obtained.

math.CA↗
arXiv2026

Lebesgue Covering Theorem and level sets of continuous functions

We formulate and prove a dimension-theoretic generalization of a version of the Lebesgue Covering Theorem. A generalized $n$-dimensional version of the Steinhaus Chessboard Theorem, recently proved algorithmically by Turzański and Ziajor, is a particular case of this result. Moreover, we study two types of sets associated with a continuous function $g \colon [0,1]^n \to \mathbb{R}$. Namely, the set of all points $p \in \mathbb{R}$ such that the fiber $g^{-1}[\left\{p\right\}]$ connects $i$th opposite faces of $[0,1]^n$, and the set of all points $p \in \mathbb{R}$ such that the fiber $g^{-1}[\left\{p\right\}]$ separates $i$th opposite faces of $[0, 1]^n$.

math.CA↗