Brunn-Minkowski type estimates for certain discrete sumsets
| Autor: | Bruch, Albert Lopez, Jing, Yifan, Mudgal, Akshat |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | Let $d,k$ be natural numbers and let $\mathcal{L}_1, \dots, \mathcal{L}_k \in \mathrm{GL}_d(\mathbb{Q})$ be linear transformations such that there are no non-trivial subspaces $U, V \subseteq \mathbb{Q}^d$ of the same dimension satisfying $\mathcal{L}_i(U) \subseteq V$ for every $1 \leq i \leq k$. For every non-empty, finite set $A \subset \mathbb{R}^d$, we prove that \[ |\mathcal{L}_1(A) + \dots + \mathcal{L}_k(A) | \geq k^d |A| - O_{d,k}(|A|^{1- \delta}), \] where $\delta >0$ is some absolute constant depending on $d,k$. Building on work of Conlon-Lim, we can show stronger lower bounds when $k$ is even and $\mathcal{L}_1, \dots, \mathcal{L}_k$ satisfy some further incongruence conditions, consequently resolving various cases of a conjecture of Bukh. Moreover, given any $d, k\in \mathbb{N}$ and any finite, non-empty set $A \subset \mathbb{R}^d$ not contained in a translate of some hyperplane, we prove sharp lower bounds for the cardinality of the $k$-fold sumset $kA$ in terms of $d,k$ and $|A|$. This can be seen as a $k$-fold generalisation of Freiman's lemma. Comment: 17 pages |
| Databáze: | arXiv |
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