The Compression method and applications

TL;DR

This paper introduces a novel method for compressing points in space, defining concepts like mass, rank, entropy, cover, and energy, and applies it to prove inequalities related to Diophantine equations.

Contribution

It develops a new compression framework with associated measures and demonstrates its application to deriving inequalities in number theory.

Findings

Established bounds for products of reciprocals of natural numbers.

Proved existence of point sets satisfying specific inequalities.

Linked compression concepts to Diophantine inequality proofs.

Abstract

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}