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A variational principle in the parametric geometry of numbers.(English)Zbl 1553.11065

We extend the parametric geometry of numbers (initiated byW. M. Schmidt andL. Summerer [Monatsh. Math. 169, No. 1, 51–104 (2013;Zbl 1264.11056)], and deepened byD. Roy [Ann. Math. (2) 182, No. 2, 739–786 (2015;Zbl 1328.11076)]) 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 of our variational principle 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 oftemplates, which generalize Roy’srigid systems. We use our variational principle to compute the Hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, resolving a conjecture ofS. Kadyrov et al. [J. Anal. Math. 133, 253–277 (2017;Zbl 1385.37005)], as well as a question ofY. Bugeaud et al. [Math. Proc. Camb. Philos. Soc. 167, No. 2, 249–284 (2019;Zbl 1450.11081)]. As a corollary of Dani’s correspondence principle [S. G. Dani, J. Reine Angew. Math. 359, 55–89 (1985;Zbl 0578.22012)], the divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs have equal Hausdorff and packing dimensions. Other applications include quantitative strengthenings of theorems due toY. Cheung [Ann. Math. (2) 173, No. 1, 127–167 (2011;Zbl 1241.11075)] andN. G. Moshchevitin [J. Lond. Math. Soc., II. Ser. 86, No. 1, 129–151 (2012;Zbl 1350.11073)], which originally resolved conjectures due toA. N. Starkov [Dynamical systems on homogeneous spaces. Providence, RI: American Mathematical Society (AMS) (1999;Zbl 1143.37300)] andW. M. Schmidt [Prog. Math. 31, 271–287 (1983;Zbl 0529.10032)]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 (1977–2016).
See [Math. Proc. Camb. Philos. Soc. 167, No. 2, 249–284 (2019;Zbl 1450.11081)] for a detailed history of the prior results.

MSC:

11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
11J13 Simultaneous homogeneous approximation, linear forms
28A80 Fractals
28A78 Hausdorff and packing measures
37A15 General groups of measure-preserving transformations and dynamical systems
37A17 Homogeneous flows
91A05 2-person games
91A44 Games involving topology, set theory, or logic

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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