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

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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