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A variational principle in the parametric geometry of numbers

Tushar Das, Lior Fishman, David Simmons, Mariusz Urbański

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.

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 linear forms in 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.

Paper Structure

This paper contains 62 sections, 43 theorems, 522 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Readers' Guide
  3. Conventions and Glossary of Notation
  4. Statements of Main results
  5. Dani correspondence
  6. Dimensions of very singular matrices
  7. Trivially singular matrices
  8. $1\times 2$ and $2\times 1$ matrices
  9. Singularity on average
  10. Starkov's conjecture
  11. Schmidt's conjecture
  12. A conjecture of BGMRV
  13. New proofs of old results
  14. The variational principle
  15. Successive minima functions and templates
  16. ...and 47 more sections

Key Result

Theorem 3.1

For all $(m,n)\neq (1,1)$, we have where ${\dim_H}(S)$ and ${\dim_P}(S)$ denote the Hausdorff and packing dimensions of a set $S$, respectively.

Figures (17)

  • Figure 1: Graphs of the dimension functions $f_1(\tau) {\, \stackrel{\mathrm{def}}{=}\, } {\dim_P}(\text{Sing}_{1,2}(\omega)) ~\text{and}~ f_2(\tau) {\, \stackrel{\mathrm{def}}{=}\, } {\dim_H}(\text{Sing}_{1,2}(\omega)).$The packing dimension function $f_1$ is linear on the intervals $[0,1/8]$ and $[1/8,1/2]$, while the Hausdorff dimension function $f_2$ is real-analytic on the intervals $[0,\tau_0]$ and $[\tau_0,1/2]$, where $\tau_0 = (3\sqrt2-2)/14 \sim 0.1602$.
  • Figure 2: The joint graph of a $1\times 2$ partial template $\mathbf f = (f_1,f_2,f_3)$, where the joint graph of a template is the union of the graphs of its component functions.
  • Figure 3: The joint graph in Figure \ref{['figuretemplate1']}, with an illustration of the sets $S_\pm(\mathbf f,I)$ and the contraction rates $\delta(\mathbf f,I)$ for each interval of linearity $I$. The "one-dimensional physics" interpretation of templates can be seen in this picture as follows: first one particle is going up while two are going down; then the top two collide into each other and their new velocity is determined by conservation of momentum; then they split apart again. Given this interpretation of the motion occurring in $I$ as being the result of "collisions" between $m$ particles going up and $n$ particles going down, $\delta(\mathbf f,I)$ counts the number of particle pairs that are "moving towards" each other (including particles "colliding" with each other).
  • Figure 4: The joint graph of a partial template $\mathbf f$ such that $S_+(\mathbf f,I_1) = \{2,\ldots,m+1\}$ and $S_+(\mathbf f,I_2) = \{1,\ldots,m\}$, where $I_1 = (t_0,t_1)$ and $I_2 = (t_1,t_2)$. In this picture we have $\mathbf f(t_0) = \mathbf f(t_2) = \mathbf 0$, and thus $|I_1| = \tfrac{n}{m+n}|I|$ and $|I_2| = \tfrac{m}{m+n}|I|$, where $I = (t_0,t_2)$. Consequently, $\frac{1}{|I|} \int_I \delta(\mathbf f,t) \;\mathrm{d} t = \frac{n}{m+n}(mn-m) + \frac{m}{m+n}(mn) = \delta_{m,n}$i.e. the average contraction rate of $\mathbf f$ over $I$ is $\delta_{m,n}$. Note that this partial template is exactly the standard template defined by the points $(t_0,0)$ and $(t_2,0)$ (cf. Definition \ref{['definitionstandardtemplate']}).
  • Figure 5: The joint graph of a template $\mathbf f$ designed to be a singular template of large Hausdorff dimension. The gray regions represent intervals where the precise value of the template is irrelevant; what matters is that the template stays away from $\mathbf 0$ on these regions.
  • ...and 12 more figures

Theorems & Definitions (118)

  • Theorem 3.1
  • Theorem 3.2: Dani4
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 3.7
  • Theorem 3.8
  • Remark
  • ...and 108 more