Counting Co-Cyclic Lattices

Abstract : There is a well-known asymptotic formula, due to W. M. Schmidt [Duke Math. J., 35 (1968), pp. 327--339], for the number of full-rank integer lattices of index at most $V$ in ${\mathbb{Z}}^n$. This set of lattices $L$ can naturally be partitioned with respect to the factor group ${\mathbb{Z}}^n/L$. Accordingly, we count the number of full-rank integer lattices $L \subseteq {\mathbb{Z}}^n$ such that ${\mathbb{Z}}^n/L$ is cyclic and of order at most $V$, and deduce that these co-cyclic lattices are dominant among all integer lattices: their natural density is $(\zeta(6) \prod_{k=4}^n \zeta(k))^{-1} \approx 85\%$. The problem is motivated by complexity theory, namely worst-case to average-case reductions for lattice problems.
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https://hal.inria.fr/hal-01382385
Contributor : Phong Q. Nguyen <>
Submitted on : Monday, October 17, 2016 - 7:01:47 AM
Last modification on : Wednesday, May 15, 2019 - 3:43:40 AM

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Phong Q. Nguyen, Igor E. Shparlinski. Counting Co-Cyclic Lattices. Siam Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2016, 30 (3), pp.1358-1370. ⟨10.1137/15M103950X⟩. ⟨hal-01382385⟩

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