On Diophantine approximations with positive integers: a remark to W.M.Schmidt's theorem

TL;DR

This paper generalizes W.M. Schmidt's theorem on Diophantine approximations, focusing on linear forms involving positive integers, and provides new insights into approximation bounds.

Contribution

It introduces a broader theorem extending Schmidt's results to linear forms with positive integer variables, enhancing understanding of Diophantine approximations.

Findings

Generalization of Schmidt's theorem for positive integer variables

New bounds for linear form approximations

Enhanced understanding of Diophantine approximation with positivity constraints

Abstract

We prove a generalization of W.M. Schmidt's theorem related to the Diophantine approximations for a linear form of the type $\alpha_1x_1+\alpha_2x_2 +y$ with {\it positive} integers $x_1,x_2$.