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Abstract
There is a wellknown asymptotic formula, due to W. M. Schmidt [Duke Math. J., 35 (1968), pp. 327339], for the number of fullrank integer lattices of index at most V in ℤ^{n}. This set of lattices L can naturally be partitioned with respect to the factor group ℤ^{n}/L. Accordingly, we count the number of fullrank integer lattices L ⊆ ℤ^{n} such that ℤ^{n}/L is cyclic and of order at most V , and deduce that these cocyclic lattices are dominant among all integer lattices: their natural density is (ς(6)II^{n}_{k=4} ς(k))^{1}≈85%. The problem is motivated by complexity theory, namely worstcase to averagecase reductions for lattice problems.
Original language  English 

Pages (fromto)  13581370 
Number of pages  13 
Journal  SIAM Journal on Discrete Mathematics 
Volume  30 
Issue number  3 
DOIs  
Publication status  Published  2016 
Externally published  Yes 
Keywords
 Cyclic lattices
 Homogeneous congruences
 Multiplicative functions
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Projects
 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
Project: Research