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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.04486 |
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| _version_ | 1866910410692624384 |
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| author | Becker, Lars Ivanisvili, Paata Krachun, Dmitry Madrid, Jóse |
| author_facet | Becker, Lars Ivanisvili, Paata Krachun, Dmitry Madrid, Jóse |
| contents | We show that for all $A, B \subseteq \{0,1,2\}^{d}$ we have $$ |A+B|\geq (|A||B|)^{\log(5)/(2\log(3))}. $$ We also show that for all finite $A,B \subset \mathbb{Z}^{d}$, and any $V \subseteq\{0,1\}^{d}$ the inequality $$ |A+B+V|\geq |A|^{1/p}|B|^{1/q}|V|^{\log_{2}(p^{1/p}q^{1/q})} $$ holds for all $p \in (1, \infty)$, where $q=\frac{p}{p-1}$ is the conjugate exponent of $p$. All the estimates are dimension free with the best possible exponents. We discuss applications to various related problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04486 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discrete Brunn-Minkowski Inequality for subsets of the cube Becker, Lars Ivanisvili, Paata Krachun, Dmitry Madrid, Jóse Combinatorics 11B30, 11B13 We show that for all $A, B \subseteq \{0,1,2\}^{d}$ we have $$ |A+B|\geq (|A||B|)^{\log(5)/(2\log(3))}. $$ We also show that for all finite $A,B \subset \mathbb{Z}^{d}$, and any $V \subseteq\{0,1\}^{d}$ the inequality $$ |A+B+V|\geq |A|^{1/p}|B|^{1/q}|V|^{\log_{2}(p^{1/p}q^{1/q})} $$ holds for all $p \in (1, \infty)$, where $q=\frac{p}{p-1}$ is the conjugate exponent of $p$. All the estimates are dimension free with the best possible exponents. We discuss applications to various related problems. |
| title | Discrete Brunn-Minkowski Inequality for subsets of the cube |
| topic | Combinatorics 11B30, 11B13 |
| url | https://arxiv.org/abs/2404.04486 |