A note on transverse sets and bilinear varieties
| Autor: | Milićević, Luka |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 11 pages |
| Databáze: | arXiv |
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