On subsets of lattice cubes avoiding affine and spherical degeneracies
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914027778605056 |
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| author | Ghosal, Anubhab Goenka, Ritesh Keevash, Peter |
| author_facet | Ghosal, Anubhab Goenka, Ritesh Keevash, Peter |
| contents | For integers $1 < k < d-1$ and $r \ge k+2$, we establish new lower bounds on the maximum number of points in $[n]^d$ such that no $r$ lie in a $k$-dimensional affine (or linear) subspace. These bounds improve on earlier results of Sudakov-Tomon and Lefmann. Further, we provide a randomised construction for the no-four-on-a-circle problem posed by Erdős and Purdy, improving Thiele's bound. We also consider the random construction in higher dimensions, and improve the bound of Suk and White for $d \geq 4$. In each case, we apply the deletion method, using results from number theory and incidence geometry to solve the associated counting problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06935 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On subsets of lattice cubes avoiding affine and spherical degeneracies Ghosal, Anubhab Goenka, Ritesh Keevash, Peter Combinatorics 05D40, 52C10, 52C35 For integers $1 < k < d-1$ and $r \ge k+2$, we establish new lower bounds on the maximum number of points in $[n]^d$ such that no $r$ lie in a $k$-dimensional affine (or linear) subspace. These bounds improve on earlier results of Sudakov-Tomon and Lefmann. Further, we provide a randomised construction for the no-four-on-a-circle problem posed by Erdős and Purdy, improving Thiele's bound. We also consider the random construction in higher dimensions, and improve the bound of Suk and White for $d \geq 4$. In each case, we apply the deletion method, using results from number theory and incidence geometry to solve the associated counting problems. |
| title | On subsets of lattice cubes avoiding affine and spherical degeneracies |
| topic | Combinatorics 05D40, 52C10, 52C35 |
| url | https://arxiv.org/abs/2509.06935 |