The asymptotic distribution of Frobenius numbers

Abstract

Abstract. The Frobenius number F (a) of an integer vector a with positive coprime coefficients is defined as the largest number that does not have a representation as a positive integer linear combination of the coefficients of a. We show that if a is taken to be random in an expanding d-dimensional domain, then F (a) has a limit distribution, which is given by the probability distribution for the covering radius of a certain simplex with respect to a (d − 1)-dimensional random lattice. This result extends recent studies for d = 3 by Arnold, Bourgain-Sinai and Shur-Sinai-Ustinov. The key features of our approach are (a) a novel interpretation of the Frobenius number in terms of the dynamics of a certain group action on the space of d-dimensional lattices, and (b) an equidistribution theorem for a multidimensional Farey sequence on closed horospheres.

Effective limit distribution of the Frobenius numbers

“…That is, the set E R = {L ∈ Ω 0 : Q 0 (L) = R} is contained in a union of finitely many bounded connected submanifolds of Ω 0 of strictly lower dimension. In fact the statement is "almost" established in [M10,Lemma 7], and we will provide some further explanation of the results there. In view of Remark 4.11, our Theorem 1.4 can be deduced with essentially the same argument as in Proposition 4.12 .…”

“…In the proof of [M10,Lemma 7], it is shown that (4.15) (4.14) ⊆ {A = (a ij ) ∈ G 0 : tr(LA) = R} , where L is the (n − 1) × (n − 1) matrix whose i-th column is n i − n d . Because L R is a relatively compact , there is a constant C R > 0, so that (4.15) can be refined to (4.16) (4.14) ⊆ {A = (a ij ) ∈ G 0 : |a ij | < C R , tr(LA) = R} .…”

Distribution of shapes of orthogonal lattices

“…The hardest case of this problem concerns the case d = 3 and was resolved by Duke [Duk88] (building on a breakthrough of Iwaniec [Iwa87]). Following Maass [Maa56,Maa59] and W. Schmidt [Sch98] (see also [Mar10] Finally we note k v SO d−1 (R) is uniquely defined by the above requirement on k v and hence [Λ v ] is canonically attached to the vector v. We refer to [Λ v ] as the shape of the lattice associated to v.…”

Feasibility of Integer Knapsacks

“…The limiting distribution of F(a) in the 3-dimensional case was derived in Shur, Sinai, and Ustinov [38], and for the general case, see Marklof [30]. Upper bounds for the average value of F(a) have been obtained in Aliev and Henk [5] and Aliev, Henk and Hinrichs [6].…”