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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.05638 |
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| _version_ | 1866929492625195008 |
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| author | Bruch, Albert Lopez Jing, Yifan Mudgal, Akshat |
| author_facet | Bruch, Albert Lopez Jing, Yifan Mudgal, Akshat |
| contents | 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 \[ |\mathcal{L}_1(A) + \dots + \mathcal{L}_k(A) | \geq k^d |A| - O_{d,k}(|A|^{1- δ}), \] where $δ>0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05638 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Brunn-Minkowski type estimates for certain discrete sumsets Bruch, Albert Lopez Jing, Yifan Mudgal, Akshat Combinatorics 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 \[ |\mathcal{L}_1(A) + \dots + \mathcal{L}_k(A) | \geq k^d |A| - O_{d,k}(|A|^{1- δ}), \] where $δ>0$ 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. |
| title | Brunn-Minkowski type estimates for certain discrete sumsets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.05638 |