The Compression method and applications
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2019
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908909859504128 |
|---|---|
| author | Agama, Theophilus |
| author_facet | Agama, Theophilus |
| contents | In this paper, we introduce and develop the method of compression of points in space. We introduce the notion of the mass, the rank, the entropy, the cover and the energy of compression. We leverage this method to prove some class of inequalities related to Diophantine equations. In particular, we show that for each $L<n-1$ and for each $K>n-1$, there exist some $(x_1,x_2,\ldots,x_n)\in \mathbb{N}^n$ with $x_i\neq x_j$ for all $1\leq i<j\leq n$ such that \begin{align} \frac{1}{K^{n}}\ll \prod \limits_{j=1}^{n}\frac{1}{x_j}\ll \frac{\log (\frac{n}{L})}{nL^{n-1}}\nonumber \end{align} and that for each $L>n-1$ there exist some $(x_1,x_2,\ldots,x_n)$ with $x_i\neq x_j$ for all $1\leq i<j\leq n$ and some $s\geq 2$ such that \begin{align} \sum \limits_{j=1}^{n}\frac{1}{x_j^s}\gg s\frac{n}{L^{s-1}}.\nonumber \end{align} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_08075 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The Compression method and applications Agama, Theophilus Number Theory 52C10, 52C35, 11H06, 52A40, 52B20 In this paper, we introduce and develop the method of compression of points in space. We introduce the notion of the mass, the rank, the entropy, the cover and the energy of compression. We leverage this method to prove some class of inequalities related to Diophantine equations. In particular, we show that for each $L<n-1$ and for each $K>n-1$, there exist some $(x_1,x_2,\ldots,x_n)\in \mathbb{N}^n$ with $x_i\neq x_j$ for all $1\leq i<j\leq n$ such that \begin{align} \frac{1}{K^{n}}\ll \prod \limits_{j=1}^{n}\frac{1}{x_j}\ll \frac{\log (\frac{n}{L})}{nL^{n-1}}\nonumber \end{align} and that for each $L>n-1$ there exist some $(x_1,x_2,\ldots,x_n)$ with $x_i\neq x_j$ for all $1\leq i<j\leq n$ and some $s\geq 2$ such that \begin{align} \sum \limits_{j=1}^{n}\frac{1}{x_j^s}\gg s\frac{n}{L^{s-1}}.\nonumber \end{align} |
| title | The Compression method and applications |
| topic | Number Theory 52C10, 52C35, 11H06, 52A40, 52B20 |
| url | https://arxiv.org/abs/1912.08075 |