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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2409.12579 |
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| _version_ | 1866918060858802176 |
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| author | Beker, Adrian Crmarić, Tonći Kovač, Vjekoslav |
| author_facet | Beker, Adrian Crmarić, Tonći Kovač, Vjekoslav |
| contents | We study optimal dimensionless inequalities $$ \|f\|_{U^k} \leq \|f\|_{\ell^{p_{k,n}}} $$ that hold for all functions $f\colon\mathbb{Z}^d\to\mathbb{C}$ supported in $\{0,1,\ldots,n-1\}^d$ and estimates $$ \|1_A\|_{U^k}^{2^k}\leq |A|^{t_{k,n}} $$ that hold for all subsets $A$ of the same discrete cubes. A general theory, analogous to the work of de Dios Pont, Greenfeld, Ivanisvili, and Madrid, is developed to show that the critical exponents are related by $p_{k,n} t_{k,n} = 2^k$. This is used to prove the three main results of the paper: an explicit formula for $t_{k,2}$, which generalizes a theorem by Kane and Tao, two-sided asymptotic estimates for $t_{k,n}$ as $n\to\infty$ for a fixed $k\geq2$, which generalize a theorem by Shao, and a precise asymptotic formula for $t_{k,n}$ as $k\to\infty$ for a fixed $n\geq2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12579 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp estimates for Gowers norms on discrete cubes Beker, Adrian Crmarić, Tonći Kovač, Vjekoslav Combinatorics Information Theory Classical Analysis and ODEs We study optimal dimensionless inequalities $$ \|f\|_{U^k} \leq \|f\|_{\ell^{p_{k,n}}} $$ that hold for all functions $f\colon\mathbb{Z}^d\to\mathbb{C}$ supported in $\{0,1,\ldots,n-1\}^d$ and estimates $$ \|1_A\|_{U^k}^{2^k}\leq |A|^{t_{k,n}} $$ that hold for all subsets $A$ of the same discrete cubes. A general theory, analogous to the work of de Dios Pont, Greenfeld, Ivanisvili, and Madrid, is developed to show that the critical exponents are related by $p_{k,n} t_{k,n} = 2^k$. This is used to prove the three main results of the paper: an explicit formula for $t_{k,2}$, which generalizes a theorem by Kane and Tao, two-sided asymptotic estimates for $t_{k,n}$ as $n\to\infty$ for a fixed $k\geq2$, which generalize a theorem by Shao, and a precise asymptotic formula for $t_{k,n}$ as $k\to\infty$ for a fixed $n\geq2$. |
| title | Sharp estimates for Gowers norms on discrete cubes |
| topic | Combinatorics Information Theory Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2409.12579 |