Equidistribution of primitive lattices in $\mathbb{R}^n$

Abstract

We count primitive lattices of rank d inside Z n , as their covolume tends to infinity, w.r.t. certain parameters of such lattices. These parameters include, for example, the direction of a lattice, which is the subsapce that it spans; its homothety class; and its equivalence class modulo rescaling and rotation, often referred to as a shape. We add to a prior work of Schmidt by allowing sets that are general enough in the spaces of parameters to conclude joint equidistribution of these parameters.The main novelty is that, in addition to the primitive d-lattices Λ themselves, we also consider their orthogonal lattices Λ ⊥ and the quotients Z n /Λ. We are able to count not only w.r.t. parameters of the primitive lattices Λ, but also w.r.t. the same parameters of Λ ⊥ and Z n /Λ. By doing so, we achieve joint equidistribution of, e.g., the shapes of primitive lattices, and the shapes of their orthogonal lattices. Finally, our asymptotic formulas for the number of primitive lattices includes an explicit error term.

Counting flags of primitive lattices

“…where vol 1 Gr 0 (d,n) is the uniform probability measure on Gr 0 (d, n). In [HK20a], we were able to extend the above result to subsets E and Φ that were general enough to conclude equidistribution, as well as to consider the orthogonal lattices…”

“…Our goal in the present paper is to generalize the counting result in [HK20a] from primitive lattices to flags of such. To this end, for d = (d 1 , .…”

“…The refinement of Franke, Manin and Tschinkel's result suggested in Theorem 1.2 could prove useful in further study of rational points on flag varieties: Browning, the first author and Wilsch [BHW21] built on [HK20a] to establish the freeness variant of Manin's conjecture, proposed by Peyre [Pey17,Pey18], for Grassmannians. We expect that Theorem 1.2 could be used to extend the results in [BHW21] from Grassmannians to more general flag varieties.…”

Equidistribution of rational subspaces and their shapes

“…Maass [Maa56,Maa59] in the 60's and Schmidt [Sch98] in the 90's have considered problems of this kind. They prove that the set of pairs (L, [L(Z)]) equidistributes in Gr n,k (R) × S k where L ∈ Gr n,k (Q) varies over the rational subspaces with discriminant at most D. In this averaged setup, Horesh and Karasik [HK20] recently verified Conjecture 1.1. Indeed, their version is polynomially effective in D.…”

Equidistribution and freeness on Grassmannians

“…The explicit equidistribution statement is given below in Theorem 5.7 with a variant phrased in terms of probability measures in Theorem 5.8. The proof takes place in Section 5 and involves a reduction to an equidistribution statement about pairs of lattices (Λ, Λ π ), for primitive Λ ∈ L m,n , that appears in recent work of Horesh and Karasik [10,Thm. 1 (4)].…”