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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.18064 |
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| _version_ | 1866911280860758016 |
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| author | esz, Szil\' ard Gy. R\' ev\' Ruzsa, Imre Z. |
| author_facet | esz, Szil\' ard Gy. R\' ev\' Ruzsa, Imre Z. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18064 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Densitometria I. Discrete groups esz, Szil\' ard Gy. R\' ev\' Ruzsa, Imre Z. Classical Analysis and ODEs Primary 22B05, Secondary 22B99, 05B10 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. |
| title | Densitometria I. Discrete groups |
| topic | Classical Analysis and ODEs Primary 22B05, Secondary 22B99, 05B10 |
| url | https://arxiv.org/abs/2511.18064 |