[Paper Review] Equidistribution of primitive vectors, and the shortest solutions to their GCD equations

Tal Horesh, Yakov Karasik|arXiv (Cornell University)|Mar 4, 2019

Advanced Algebra and Geometry24 references4 citations

TL;DR

This paper establishes effective joint equidistribution of primitive vectors in ℤⁿ by showing that their directions, orthogonal lattices, and normalized shortest solutions to gcd equations equidistribute as the vector norm tends to infinity. The key result is that the normalized solution lengths equidistribute in [0,1] with respect to a non-Lebesgue measure when n ≥ 3, resolving a higher-dimensional analog of Risager and Rudnick's equidistribution result for n=2.

ABSTRACT

We prove effective joint equidistribution of several natural parameters associated to primitive vectors in $\mathbb{Z}^{n}$, as the norm of these vectors tends to infinity. These parameters include the direction, the orthogonal lattice, and the length of the shortest solution to the associated $\gcd$ equation. We show that the first two parameters equidistribute w.r.t. the Haar measure on the corresponding spaces, which are the unit sphere and the space of unimodular rank $n-1$ lattices in $\mathbb{R}^{n}$ respectively. The main novelty is the equidistribution of the shortest solutions to the $\gcd$ equations: we show that, when normalized by the covering radius of the orthogonal lattice, the lengths of these solutions equidistribute in the interval $\left[0,1 ight]$ w.r.t. a measure that is Lebesgue only when $n=2$, and non-Lebesgue otherwise. These equidistribution results are deduced from effectively counting lattice points in domains which are defined w.r.t. a generalization of the Iwasawa decomposition in simple algebraic Lie groups, where we apply a method due to A. Gorodnik and A. Nevo.

Motivation & Objective

• To resolve the higher-dimensional analog of Risager and Rudnick's equidistribution result for gcd solutions in ℤⁿ when n ≥ 3.
• To identify the correct normalization for shortest solutions to gcd equations in dimensions n ≥ 3, since ‖w_v‖/‖v‖ fails to equidistribute.
• To determine the limiting measure on [0,1] for normalized shortest solution lengths, showing it is Lebesgue only when n=2.
• To establish effective equidistribution of directions and shapes of orthogonal lattices using methods from homogeneous dynamics.
• To develop a counting framework for lattice points in Lie groups using refined Iwasawa decomposition and well-rounded families.

Proposed method

• Applies a method of Gorodnik and Nevo for effective lattice point counting in well-rounded families of sets in SLₙ(ℝ).
• Uses a refined Iwasawa decomposition of SLₙ(ℝ) to parameterize matrices and define fundamental domains for lattices and directions.
• Introduces the Iwasawa roundomorphism to relate the geometry of solutions to the structure of the group SLₙ(ℝ).
• Defines base sets and families of domains (e.g., Ω_T^S(Ψ)) that capture the relevant arithmetic and geometric parameters.
• Establishes that the counting domain is both LWR (locally well-rounded) and BLC (bounded Lipschitz constant) to apply effective equidistribution theorems.
• Applies Theorem 8.4 on lattice point counting with error terms depending on spectral gap τ(Γ) and dimension.

Experimental results

Research questions

• RQ1 What is the correct normalization for the shortest solution to the gcd equation a₁x₁ + ⋯ + aₙxₙ = 1 in dimensions n ≥ 3?
• RQ2 In which interval do the normalized shortest solutions fall, and with respect to which measure do they equidistribute?
• RQ3 Do the directions and shapes of orthogonal lattices to primitive vectors equidistribute in their respective spaces?
• RQ4 Can effective equidistribution results be proven for all three parameters—direction, orthogonal lattice shape, and normalized solution length—simultaneously?
• RQ5 What is the rate of equidistribution, and how does it depend on the spectral gap τ(Γ) of the lattice?

Key findings

• The normalized shortest solution lengths equidistribute in [0,1] with respect to a measure that is Lebesgue only when n=2, and non-Lebesgue otherwise.
• The set of primitive vectors for which ‖w_v‖/‖v‖ → 0 has full density in ℤⁿ_prim as ‖v‖ → ∞, showing that normalization by ‖v‖ is ineffective for n ≥ 3.
• Directions of primitive vectors equidistribute with respect to the Haar measure on the unit sphere Sⁿ⁻¹.
• Shapes of orthogonal lattices Λ_v equidistribute with respect to the invariant measure on the space SO_{n−1}(ℝ)\SL_{n−1}(ℝ)/SL_{n−1}(ℤ).
• The counting error term in the equidistribution result is O_Ψ,ε(e^{nT(1−τ+δ+ε)}), showing effective convergence with explicit dependence on the spectral gap τ(Γ).
• The main term in the counting formula is asymptotically proportional to μ(Ω_T^S(Ψ))/μ(G/Γ), with the error bounded by a sub-exponential factor in T.