  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.

Number Theory