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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 ℤn. This set of lattices L can naturally be partitioned with respect to the factor group ℤn/L. Accordingly, we count the number of full-rank integer lattices L ⊆ ℤn such that ℤn/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 (ς(6)IInk=4 ς(k))-1≈85%. The problem is motivated by complexity theory, namely worst-case to average-case reductions for lattice problems.
|Number of pages||13|
|Journal||SIAM Journal on Discrete Mathematics|
|Publication status||Published - 2016|
- Cyclic lattices
- Homogeneous congruences
- Multiplicative functions
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- 1 Finished
New Applications of Additive Combinatorics in Number Theory and Graph Theory
Mans, B., Shparlinski, I., MQRES, M. & PhD Contribution (ARC), P. C.
1/01/14 → 31/12/17