Diophantine approximation with sign constraints

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, 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 $2$ , Moshchevitin proved very recently that it cannot be replaced by a number larger than $1.947$ . In this paper, we present a construction of points $(1,alpha ,beta )$ showing that the result of Schmidt is in fact optimal. These points also possess strong additional Diophantine properties that are described in the paper.