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An asymptotic formula is found for the average number of local minima of three-dimensional complete integral lattices with determinant in the interval [1, N ]. This is a generalization to the two-dimensional case of the classical result about the average length of a finite continued fraction with denominator belonging to [1, N ]. Notation R+ is the set of positive real numbers; Xs×s is the set of (s× s)-matrices with entries in X; GLs(R) is the set of nonsingular matrices in R s×s; GLs(X;N) = {M ∈ Xs×s : detM = N}; for m, . . . ,m ∈ R, the symbol [m, . . . ,m] denotes the matrix with the columns m T , . . . ,m T ; ∂Ω is the boundary of a set Ω, and s Ω is the closure of Ω; #X is the cardinality of a finite set X. We write f(x) g(x) (or f(x) = O(g(x))) for x ∈ X if there exists a constant C > 0 such that |f(x)| ≤ C · g(x) for all x ∈ X. If C depends on a parameter θ, we write f(x) θ g(x) (or f(x) = Oθ(g(x))). We write f g if f g f . Introduction Two generalizations of continued fractions to the multidimensional case emerged at the end of the 19th century. One of them was due to F. Klein [1], the other was suggested by Voronŏı [2], and, independently, by Minkowski [3]. Each one is related to isolation and the study of certain sets of nodes of an s-dimensional lattice Γ, specifically, the Klein polytops and (accordingly) the set M(Γ) of all local minima for Γ. We recall the definitions. A complete s-dimensional integral lattice is a set of the form

Average number of local minima for three-dimensional integral lattices