[Paper Review] Spherical averages of Siegel transforms for higher rank diagonal actions and applications

Jayadev S. Athreya, Anish Ghosh+1 more|arXiv (Cornell University)|Jul 14, 2014

Advanced Algebra and Geometry10 references6 citations

TL;DR

This paper establishes a novel higher-rank averaging result for Siegel transforms on the space of unimodular lattices in $\mathbb{R}^n$, leveraging spherical averages to derive equidistribution results for approximates in multiplicative and weighted Diophantine approximation. The key contribution is a general equidistribution theorem for spherical averages under higher-rank diagonal actions, extending prior work on Dirichlet's theorem and enabling new quantitative results on the geometry of approximates.

ABSTRACT

We investigate the geometry of approximates in multiplicative Diophantine approximation. Our main tool is a new multiparameter averaging result for Siegel transforms on the space of unimodular lattices in ${\mathbb R}^n$ which is of independent interest.

Motivation & Objective

• To extend equidistribution results for approximates in Diophantine approximation beyond the classical Dirichlet setting to multiplicative and weighted variants.
• To develop a new averaging technique for Siegel transforms under higher-rank diagonal actions on $\mathrm{SL}_n(\mathbb{R})/\mathrm{SL}_n(\mathbb{Z})$.
• To establish a general equidistribution result for spherical averages of Siegel transforms that applies to higher-rank diagonal flows.
• To apply this result to analyze the geometric distribution of approximates in multiplicative and weighted Diophantine approximation problems.
• To generalize previous results on spiraling approximates to higher-rank settings and more general Diophantine inequalities.

Proposed method

• The authors introduce a new averaging method over compact subgroups $K$ of $\mathrm{SO}_n(\mathbb{R})$ to study spherical averages of Siegel transforms on the space of unimodular lattices.
• They use the Siegel transform $\widehat{f}(\Lambda) = \sum_{\mathbf{v} \in \Lambda \setminus \{0\}} f(\mathbf{v})$ for bounded, Riemann-integrable functions $f$ with compact support.
• The main technical tool is a limit formula: $\lim_{t \to \infty} \int_K \widehat{f}(g_t k \Lambda) \, dk = \int_{X_{d+1}} \widehat{f} \, d\mu$, where $g_t$ is a higher-rank diagonal flow.
• The proof relies on equidistribution of spherical averages under the action of diagonal flows, extending techniques from one-parameter flows to higher-rank settings.
• The authors analyze the geometry of thin regions $R_{\epsilon,T}$ and $R_{A,\epsilon,T}$ in $\mathbb{R}^{d+1}$ that capture lattice points corresponding to Diophantine approximates.
• They use volume estimates on cones and spherical caps to control the measure of sets where the normalized directions of approximates lie, particularly near coordinate axes.

Experimental results

Research questions

• RQ1 How do the directions of Diophantine approximates distribute on the unit sphere when the approximation is multiplicative or weighted rather than uniform?
• RQ2 Can spherical averages of Siegel transforms be equidistributed under higher-rank diagonal actions on the space of unimodular lattices?
• RQ3 What is the asymptotic behavior of the number of lattice points in thin regions $R_{A,\epsilon,T}$ as $T \to \infty$?
• RQ4 To what extent can the equidistribution of approximates in Dirichlet’s theorem be generalized to number fields with multiple places, including $p$-adic completions?
• RQ5 Can the methods used for one-parameter flows be extended to higher-rank diagonal actions to yield new averaging theorems?

Key findings

• The spherical averages of Siegel transforms equidistribute under higher-rank diagonal actions: $\lim_{t \to \infty} \int_K \widehat{f}(g_t k \Lambda) \, dk = \int_{X_{d+1}} \widehat{f} \, d\mu$.
• For any unimodular lattice $\Lambda$, the normalized number of approximates in a spherical region $A \subset \mathbb{S}^{d-1}$ converges to $\mathrm{vol}(A)$ as $T \to \infty$, under the spherical average.
• The result implies that approximates in multiplicative and weighted Diophantine approximation are uniformly distributed in direction on the sphere, extending the spiraling phenomenon to higher-rank settings.
• The volume of the set $\mathcal{C}A \cap S$ grows like $\int_1^\infty \tau^{m-1} \, d\tau = \infty$ when $A$ is near a coordinate axis, indicating infinite density of approximates in such regions.
• The method applies to number fields with multiple infinite and finite places, suggesting a path to generalizing the results to $p$-adic and adelic settings.
• The proof technique avoids exponential mixing and instead uses Riemann integration and geometric volume estimates, offering a new, elementary approach to higher-rank equidistribution.