Some results on random unimodular lattices

DOI10.1090/PROC/15241zbMATH Open1459.11145arXiv1909.05205OpenAlexW3004317995MaRDI QIDQ5145508

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Publication date: 20 January 2021

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Abstract: Let $ninmathbb{Z}_{geq3}.$ Given any Borel subset $A$ of $mathbb{R}^n$ with finite and nonzero measure, we prove that the probability that the set of primitive points of a random full-rank unimodular lattice in $mathbb{R}^n$ does not contain any $mathbbR$-linearly independent subset of $A$ of cardinality $(n-2)$ is bounded from above by a constant multiple, which depends only on $n$, of $left(mathrmvol(A)ight{)}^{-1}.$ This generalizes a result that is jointly due to J. S. Athreya and G. A. Margulis (see cite[Theorem 2.2]{Log}). We also generalize independent results of C. A. Rogers (see cite[Theorem 6]{MeanRog}) and W. M. Schmidt (see cite[Theorem 1]{Metrical}) about primitive lattice points of random lattices to the case of primitive tuples of rank less than $fracn2.$ In addition to the work of the authors who were just mentioned, a crucial element of this present paper is the usage of a rearrangement inequality due to Brascamp extendash Lieb extendash Luttinger (see cite[Theorem 3.4]{BLL}).

Full work available at URL: https://arxiv.org/abs/1909.05205

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