Discretised sum-product theorems by Shannon-type inequalities
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866918207126765568 |
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| author | Máthé, András O'Regan, William |
| author_facet | Máthé, András O'Regan, William |
| contents | By making use of arithmetic information inequalities, we give a strong quantitative bound for the discretised ring theorem. In particular, we show that if $A \subset [1,2]$ is a $(δ,σ)$-set, with $|A| = δ^{-σ},$ then $A+A$ or $AA$ has $δ$-covering number at least $δ^{-c}|A|$ for any $0 < c < \min\{σ/6, (1-σ)/6\}$ provided that $δ> 0$ is small enough. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_02943 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Discretised sum-product theorems by Shannon-type inequalities Máthé, András O'Regan, William Classical Analysis and ODEs Combinatorics Metric Geometry 28A78 (Primary) 28A80, 05B99 (Secondary) By making use of arithmetic information inequalities, we give a strong quantitative bound for the discretised ring theorem. In particular, we show that if $A \subset [1,2]$ is a $(δ,σ)$-set, with $|A| = δ^{-σ},$ then $A+A$ or $AA$ has $δ$-covering number at least $δ^{-c}|A|$ for any $0 < c < \min\{σ/6, (1-σ)/6\}$ provided that $δ> 0$ is small enough. |
| title | Discretised sum-product theorems by Shannon-type inequalities |
| topic | Classical Analysis and ODEs Combinatorics Metric Geometry 28A78 (Primary) 28A80, 05B99 (Secondary) |
| url | https://arxiv.org/abs/2306.02943 |