Determine the higher-dimensional Diophantine approximation constant

Determine the optimal universal constant D_opt(d, ||·||) for simultaneous Diophantine approximation in dimension d with respect to an arbitrary norm, beyond the cases currently known, including the unresolved dimensions and norms identified in the paper.

Background

For every x in Rd, Dirichlet-type approximation guarantees infinitely many rational approximations with an error controlled by a constant D_{d,||·||}. The smallest such constant is denoted D_opt(d,||·||). The paper notes that its exact value is known in dimension one and in the two-dimensional Euclidean case, but not generally in higher dimensions.

References

In higher dimensions the best possible choice of D_{opt}(d,||·||) is unknown except for the case d=2 and ||·||{2} the Euclidean norm, where it was proven that D{opt}(2,||·||_{2})=\sqrt{\frac{2}{\sqrt{23}}}.

— Distribution of points near the origin in the $d$-dimensional Lagrange spectrum  (2609.05311 - Ward, 4 Sep 2026) in Section 1, Introduction