Diophantine approximation in angular domains

TL;DR

This paper proves that Schmidt's bound on the approximation of real numbers in positive integer triples is optimal, confirming that the exponent cannot be improved beyond a certain limit.

Contribution

The authors construct a specific example demonstrating that Schmidt's approximation result cannot be strengthened further, establishing the optimality of his bound.

Findings

Schmidt's exponent limit is proven to be sharp.

Constructive proof confirms no better exponent than Schmidt's.

Results refine understanding of Diophantine approximation in positive domains.

Abstract

Let $\alpha$ and $\beta$ be real numbers such that $1$, $\alpha$ and $\beta$ are linearly independent over $\mathbb{Q}$. A classical result of Dirichlet asserts that there are infinitely many triples of integers $(x_0,x_1,x_2)$ such that $|x_0+\alpha x_1+\beta x_2| < \max\{|x_1|,|x_2|\}^{-2}$. In 1976, W. M. Schmidt asked what can be said under the restriction that $x_1$ and $x_2$ be positive. Upon denoting by $\gamma\cong 1.618$ the golden ratio, he proved that there are triples $(x_0,x_1,x_2) \in \mathbb{Z}^3$ with $x_1,x_2>0$ for which the product $|x_0 + \alpha x_1 + \beta x_2| \max\{|x_1|,|x_2|\}^\gamma$ is arbitrarily small. Although Schmidt later conjectured that $\gamma$ can be replaced by any number smaller than $2$, N. Moshchevitin proved very recently that it cannot be replaced by a number larger than $1.947$. In this paper, we present a construction showing that the result of Schmidt is in fact optimal.