Distribution of points near the origin in the $d$-dimensional Lagrange spectrum

Abstract

We develop a new framework, inspired by Schmidt's games, to study the Lagrange spectrum for simultaneous Diophantine approximation in dimension $d\geq 2$. We show the Hausdorff dimension of the set of points in $\mathbb{R}^{d}$ whose best approximation constant lies in $[\varepsilon,\varepsilon(1+\delta\varepsilon^{d})]$, for some constant $\delta>0$, is positive, and for a slightly larger set approaches full dimension as $\varepsilon \to 0$. We also show that the box dimension of the $d$-dimensional Lagrange spectrum is bounded from below by $1-\frac{1}{d+1}$. The proof combines a novel application of the Simplex lemma near rational points with the game-theoretic framework. As an additional result we use an elementary observation to show that the naturally defined Lagrange spectrum for systems of linear forms is uncountable in the case of square matrices.