Abstract
We investigate in this paper the distribution of the discrepancy of various lattice counting functions. In particular, we prove that the number of lattice points contained in certain domains defined by products of linear forms satisfies a Central Limit Theorem. Furthermore, we show that the Central Limit Theorem holds for the number of rational approximants for weighted Diophantine approximation in ?d. Our arguments exploit chaotic properties of the Cartan flow on the space of lattices.
Highlights
Let ΩT ⊂ Rd be an increasing family of compact domains, and let Ld denote the space of lattices in Rd with covolume one, endowed with the unique SLd(R)-invariant probability measure λd
We stress that it is unlikely that a Central Limit Theorem (CLT) holds for general domains; for instance, the counting of lattice points in the regions
We prove that if some regularity is imposed on D, a suitable Central Limit Theorem holds
Summary
Let ΩT ⊂ Rd be an increasing family of compact domains, and let Ld denote the space of lattices in Rd with covolume one, endowed with the unique SLd(R)-. In [14, 15], Schmidt showed that for generic lattices Λ ∈ Ld,. +ε for all ε > 0., and the counting function Λ → |Λ ∩ ΩT | exhibits cancellations of the same order as a sum of independent random variables. The argument in [14] implicitly follows the heuristic approach outlined above and proves some form of pairwise independence using arithmetic considerations. The aim of this work is to establish a Central Limit Theorem (CLT) in this setting, at least under some additional assumptions on the domains ΩT. We stress that it is unlikely that a CLT holds for general domains; for instance, the counting of lattice points in the regions.
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