Additive bases, coset covers, and non-vanishing linear maps
| Autor: | Nagy, János, Pach, Péter Pál, Tomon, István |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Recently, the first two authors proved the Alon-Jaeger-Tarsi conjecture on non-vanishing linear maps, for large primes. We extend their ideas to address several other related conjectures. We prove the weak Additive Basis conjecture proposed by Szegedy, making a significant step towards the Additive Basis conjecture of Jaeger, Linial, Payan, and Tarsi. In fact, we prove it in a strong form: there exists a set $A\subset\mathbb{F}_p^*$ of size $O(\log p)$ such that if $B\subset\mathbb{F}_p^{n}$ is the union of $p$ linear bases, then $A\cdot B=\{a\cdot v:a\in A, v\in B\}$ is an additive basis. An old result of Tomkinson states that if $G$ is a group, and $\{H_{i}x_{i}:i\in [k]\}$ is an irredundant coset cover of $G$, then $|G:\bigcap_{i\in [k]} H_{i}|\leq k!,$ and this bound is the best possible. It is a longstanding open problem whether the upper bound can be improved to $e^{O(k)}$ in case we restrict cosets to subgroups. Pyber proposed to study this question for abelian groups. We show that somewhat surprisingly, if $G$ is abelian, the upper bound can be improved to $e^{O(k\log \log k)}$ already in the case of general coset covers, making the first substantial improvement over the $k!$ bound. Finally, we prove a natural generalization of the Alon-Jaeger-Tarsi conjecture for multiple matrices. Comment: 13 pages |
| Databáze: | arXiv |
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