Lattice Points in Large Borel Sets and Successive Minima

2005/10/08 by Iskander Aliev, Aliev, Iskander, Peter M. Gruber +2
Mathematics · #11H16 #11H50 #11J25 #Advanced Topology and Set Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT #msc:11H16 #msc:11H50 #msc:11J25

paper · pdf · doi:10.48550/arxiv.math/0510163

8 pages

arxiv created 2005/10/08 · openalex publication_date 2005/10/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let B be a Borel set in \mathbb E^d with volume V(B)=∞. It is shown that almost all lattices L in \mathbb E^d contain infinitely many pairwise disjoint d-tuples, that is sets of d linearly independent points in B. A consequence of this result is the following: let S be a star body in \mathbb E^d with V(S)=∞. Then for almost all lattices L in \mathbb E^d the successive minima λ₁(S,L),..., λ_d(S,L) of S with respect to L are 0. A corresponding result holds for most lattices in the Baire category sense. A tool for the latter result is the semi-continuity of the successive minima.

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• Successive Minima and Lattice Points
• Higher-rank pointwise discrepancy bounds and logarithm laws for generic lattices
• Reprint: On lattice points in n -dimensional star bodies (1946)
• Some measure-theoretical and dynamical properties of successive minima on the space of unimodular lattices
• Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic