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On the Poisson distribution of lengths of lattice vectors in a random lattice

Abstract

We prove that the volumes determined by the lengths of the non-zero vectors ±x in a random lattice L of covolume 1 define a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity \({\frac{1}{2}}\) . This generalizes earlier results by Rogers (Proc Lond Math Soc (3) 6:305–320, 1956, Thm. 3) and Schmidt (Acta Math 102:159–224, 1959, Satz 10).

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Correspondence to Anders Södergren.

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Södergren, A. On the Poisson distribution of lengths of lattice vectors in a random lattice. Math. Z. 269, 945–954 (2011). https://doi.org/10.1007/s00209-010-0772-8

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  • DOI: https://doi.org/10.1007/s00209-010-0772-8

Mathematics Subject Classification (2000)

  • 11H06
  • 60G55