Diophantine approximation in angular domains

Abstract

Let $α$ and $β$ be real numbers such that $1$, $α$ and $β$ 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 $|x_0+αx_1+βx_2| < \max\{|x_1|,|x_2|\}^{-2}$. In 1976, W. M. Schmidt asked what can be said under the restriction that $x_1$ and $x_2$ be positive. Upon denoting by $γ\cong 1.618$ 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 $|x_0 + αx_1 + βx_2| \max\{|x_1|,|x_2|\}^γ$ is arbitrarily small. Although Schmidt later conjectured that $γ$ 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.