A variational principle in the parametric geometry of numbers

Abstract

We extend the parametric geometry of numbers (initiated by Schmidt and Summerer, and deepened by Roy) to Diophantine approximation for systems of $m$ linear forms in $n$ variables, and establish a new connection to the metric theory via a variational principle that computes fractal dimensions of a variety of sets of number-theoretic interest. The proof relies on two novel ingredients: a variant of Schmidt's game capable of computing the Hausdorff and packing dimensions of any set, and the notion of templates, which generalize Roy's rigid systems. In particular, we compute the Hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, resolving a conjecture of Kadyrov, Kleinbock, Lindenstrauss and Margulis, as well as a question of Bugeaud, Cheung and Chevallier. As a corollary of Dani's correspondence principle, the divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs has equal Hausdorff and packing dimensions. Other applications include quantitative strengthenings of theorems due to Cheung and Moshchevitin, which originally resolved conjectures due to Starkov and Schmidt respectively; as well as dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions, completing partial results due to Baker, Bugeaud, Cheung, Chevallier, Dodson, Laurent and Rynne.

arXiv commentnow with a uniform version of the main variational principle

References in corpus (13)

• The set of badly approximable vectors is strongly $C^1$ incompressible
• On Schmidt and Summerer parametric geometry of numbers
• On Diophantine exponents and Khintchine's transference principle
• Two-dimensional badly approximable vectors and Schmidt's game
• A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation
• Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces
• Spectrum of the exponents of best rational approximation
• Hausdorff dimension and uniform exponents in dimension two
• Proof of W.M.Schmidt's conjecture concerning successive minima of a lattice
• Hausdorff dimension of divergent trajectories on homogeneous space
• Determinacy and indeterminacy of games played on complete metric spaces
• Parametric geometry of numbers in function fields
• On a Conjecture of Schmidt for the Parametric Geometry of Numbers

Cited by in corpus (10)

• Rational approximation to real points on quadratic hypersurfaces
• Divergent trajectories on products of homogeneous spaces
• Going-up theorems for simultaneous Diophantine approximation
• K-divergent lattices
• An upper bound of the Hausdorff dimension of singular vectors on affine subspaces
• Singularity, weighted uniform approximation, intersections and rates
• Parametric Geometry of Numbers with General Flow
• On Nontrivial Winning and Losing Parameters of Schmidt Games
• Good functions, measures, and the Kleinbock-Tomanov conjecture
• On the packing dimension of weighted singular matrices on fractals