Additive energies of subsets of discrete cubes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913580841959424 |
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| author | Shao, Xuancheng |
| author_facet | Shao, Xuancheng |
| contents | For a positive integer $n \geq 2$, define $t_n$ to be the smallest number such that the additive energy $E(A)$ of any subset $A \subset \{0,1,\cdots,n-1\}^d$ and any $d$ is at most $|A|^{t_n}$. Trivially we have $t_n \leq 3$ and $$ t_n \geq 3 - \log_n\frac{3n^3}{2n^3+n} $$ by considering $A = \{0,1,\cdots,n-1\}^d$. In this note, we investigate the behavior of $t_n$ for large $n$ and obtain the following non-trivial bounds: $$ 3 - (1+o_{n\rightarrow\infty}(1)) \log_n \frac{3\sqrt{3}}{4} \leq t_n \leq 3 - \log_n(1+c), $$ where $c>0$ is an absolute constant. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_06944 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Additive energies of subsets of discrete cubes Shao, Xuancheng Combinatorics Number Theory 11B30 For a positive integer $n \geq 2$, define $t_n$ to be the smallest number such that the additive energy $E(A)$ of any subset $A \subset \{0,1,\cdots,n-1\}^d$ and any $d$ is at most $|A|^{t_n}$. Trivially we have $t_n \leq 3$ and $$ t_n \geq 3 - \log_n\frac{3n^3}{2n^3+n} $$ by considering $A = \{0,1,\cdots,n-1\}^d$. In this note, we investigate the behavior of $t_n$ for large $n$ and obtain the following non-trivial bounds: $$ 3 - (1+o_{n\rightarrow\infty}(1)) \log_n \frac{3\sqrt{3}}{4} \leq t_n \leq 3 - \log_n(1+c), $$ where $c>0$ is an absolute constant. |
| title | Additive energies of subsets of discrete cubes |
| topic | Combinatorics Number Theory 11B30 |
| url | https://arxiv.org/abs/2407.06944 |