Avoiding intersections of given size in finite affine spaces AG(n,2)
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914816027787264 |
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| author | Kovács, Benedek Nagy, Zoltán Lóránt |
| author_facet | Kovács, Benedek Nagy, Zoltán Lóránt |
| contents | We study the set of intersection sizes of a k-dimensional affine subspace and a point set of size m \in [0, 2^n] of the n-dimensional binary affine space AG(n,2). Following the theme of Erdős, Füredi, Rothschild and T. Sós, we partially determine which local densities in k-dimensional affine subspaces are unavoidable in all $m$-element point sets in the n-dimensional affine space. We also show constructions of point sets for which the intersection sizes with $k$-dimensional affine subspaces takes values from a set of a small size compared to 2^k. These are built up from affine subspaces and so-called subspace evasive sets. Meanwhile, we improve the best known upper bounds on subspace evasive sets and apply results concerning the canonical signed-digit (CSD) representation of numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_05632 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Avoiding intersections of given size in finite affine spaces AG(n,2) Kovács, Benedek Nagy, Zoltán Lóránt Combinatorics We study the set of intersection sizes of a k-dimensional affine subspace and a point set of size m \in [0, 2^n] of the n-dimensional binary affine space AG(n,2). Following the theme of Erdős, Füredi, Rothschild and T. Sós, we partially determine which local densities in k-dimensional affine subspaces are unavoidable in all $m$-element point sets in the n-dimensional affine space. We also show constructions of point sets for which the intersection sizes with $k$-dimensional affine subspaces takes values from a set of a small size compared to 2^k. These are built up from affine subspaces and so-called subspace evasive sets. Meanwhile, we improve the best known upper bounds on subspace evasive sets and apply results concerning the canonical signed-digit (CSD) representation of numbers. |
| title | Avoiding intersections of given size in finite affine spaces AG(n,2) |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2305.05632 |