A note on transverse sets and bilinear varieties
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911612145762304 |
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| author | Milićević, Luka |
| author_facet | Milićević, Luka |
| contents | Let $G$ and $H$ be finite-dimensional vector spaces over $\mathbb{F}_p$. A subset $A \subseteq G \times H$ is said to be transverse if all of its rows $\{x \in G \colon (x,y) \in A\}$, $y \in H$, are subspaces of $G$ and all of its columns $\{y \in H \colon (x,y) \in A\}$, $x \in G$, are subspaces of $H$. As a corollary of a bilinear version of Bogolyubov argument, Gowers and the author proved that dense transverse sets contain bilinear varieties of bounded codimension. In this paper, we provide a direct combinatorial proof of this fact. In particular, we improve the bounds and evade the use of Fourier analysis and Freiman's theorem and its variants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_15175 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on transverse sets and bilinear varieties Milićević, Luka Combinatorics Number Theory Let $G$ and $H$ be finite-dimensional vector spaces over $\mathbb{F}_p$. A subset $A \subseteq G \times H$ is said to be transverse if all of its rows $\{x \in G \colon (x,y) \in A\}$, $y \in H$, are subspaces of $G$ and all of its columns $\{y \in H \colon (x,y) \in A\}$, $x \in G$, are subspaces of $H$. As a corollary of a bilinear version of Bogolyubov argument, Gowers and the author proved that dense transverse sets contain bilinear varieties of bounded codimension. In this paper, we provide a direct combinatorial proof of this fact. In particular, we improve the bounds and evade the use of Fourier analysis and Freiman's theorem and its variants. |
| title | A note on transverse sets and bilinear varieties |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2308.15175 |