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

We develop a new framework, inspired by Schmidt’s games, to study the Lagrange spectrum for simultaneous Diophantine approximation in dimension $d\ge 2$. We show the Hausdorff dimension of the set of points in ${\mathbb{R}}^d$ whose best approximation constant lies in $[\epsilon ,\epsilon (1+\delta {\epsilon }^d)]$, for some constant $\delta >0$, is positive, and for a slightly larger set approaches full dimension as $\epsilon \to 0$. The proof combines a novel application of the Simplex lemma near rational points with the game-theoretic framework. We also 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.