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

@article{Sdergren2010OnTP,
title={On the Poisson distribution of lengths of lattice vectors in a random lattice},
author={Anders S{\"o}dergren},
journal={Mathematische Zeitschrift},
year={2010},
volume={269},
pages={945-954}
}
• A. Södergren
• Published 20 January 2010
• Mathematics
• Mathematische Zeitschrift
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).
We use an idea from sieve theory—specifically, an inclusion–exclusion argument inspired by Schmidt (Proc Am Math Soc 9:390–403, 1958)—to estimate the distribution of the lengths of kth shortest
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