Joint upper Banach density, VC dimensions and Euclidean point configurations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914525494640640 |
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| author | Predojević, Bruno |
| author_facet | Predojević, Bruno |
| contents | We study two related quantities which generalize the concept of upper Banach density of a set to two measurable subsets of the plane. The first of them allows us to generalize a classic result on sufficiently large distances realized in a set of positive upper density, to distances between points of two sets satisfying an appropriate density condition. The second one allows us to show that for all sufficiently large scales $t>0$ and for a smooth, closed, centrally symmetric, planar curve $Γ$ which bounds a convex and compact region in the plane and is of non-vanishing curvature, the family consisting of portions of translates of $tΓ$ has the maximal possible Vapnik--Chervonenkis dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17453 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Joint upper Banach density, VC dimensions and Euclidean point configurations Predojević, Bruno Classical Analysis and ODEs Combinatorics We study two related quantities which generalize the concept of upper Banach density of a set to two measurable subsets of the plane. The first of them allows us to generalize a classic result on sufficiently large distances realized in a set of positive upper density, to distances between points of two sets satisfying an appropriate density condition. The second one allows us to show that for all sufficiently large scales $t>0$ and for a smooth, closed, centrally symmetric, planar curve $Γ$ which bounds a convex and compact region in the plane and is of non-vanishing curvature, the family consisting of portions of translates of $tΓ$ has the maximal possible Vapnik--Chervonenkis dimension. |
| title | Joint upper Banach density, VC dimensions and Euclidean point configurations |
| topic | Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2510.17453 |