Lattice Points in Large Borel Sets and Successive Minima

Let $B$ be a Borel set in $\mathbb E^{d}$ with volume $V(B)=\infty$. It is shown that almost all lattices $L$ in $\mathbb E^{d}$ contain infinitely many pairwise disjoint $d$-tuples, that is sets of $d$ linearly independent points in $B$. A consequence of this result is the following: let $S$ be a star body in $\mathbb E^{d}$ with $V(S)=\infty$. Then for almost all lattices $L$ in $\mathbb E^{d}$ the successive minima $\lambda_{1}(S,L),..., \lambda_{d}(S,L)$ of $S$ with respect to $L$ are 0. A corresponding result holds for most lattices in the Baire category sense. A tool for the latter result is the semi-continuity of the successive minima.Comment: 8 page

An Optimal Lower Bound for the Frobenius Problem

Given $N$ positive integers $a_1, ..., a_N$ with $\gcd(a_1, ..., a_N)=1$, let $f_N$ denote the largest natural number which is not a positive integer combination of $a_1, ..., a_N$. This paper gives an optimal lower bound for $f_N$ in terms of the absolute inhomogeneous minimum of the standard $(N-1)$-simplex.Comment: 10 page

Integer Knapsacks: Average Behavior of the Frobenius Numbers

The main result of the paper shows that the asymptotic growth of the Frobenius number in average is significantly slower than the growth of the maximum Frobenius number

Successive Minima and Best Simultaneous Diophantine Approximations

We study the problem of best approximations of a vector $\alpha\in{\mathbb R}^n$ by rational vectors of a lattice $\Lambda\subset {\mathbb R}^n$ whose common denominator is bounded. To this end we introduce successive minima for a periodic lattice structure and extend some classical results from geometry of numbers to this structure. This leads to bounds for the best approximation problem which generalize and improve former results.Comment: 8 page

Best Simultaneous Diophantine Approximations under a Constraint on the Denominator

We investigate the problem of best simultaneous Diophantine approximation under a constraint on the denominator, as proposed by Jurkat. New lower estimates for optimal approximation constants are given in terms of critical determinants of suitable star bodies. Tools are results on simultaneous Diophantine approximation of rationals by rationals with smaller denominator. Finally, the approximation results are applied to the decomposition of integer vectors.Comment: 17 pages, corrected typo

A sharp upper bound for the Lattice Programming Gap

Abstract. Given a full-dimensional lattice Λ ⊂ Z d and a vector l ∈ Qd >0 , we consider the family of the lattice problems Minimize {l · x : x ≡ r( mod Λ), x ∈ Z d ≥0} , r ∈ Z d (0.1) . The lattice programming gap gap(Λ,l) is the largest value of the minima in (0.1) as r varies over Z d . We obtain a sharp upper bound for gap(Λ,l)

On polynomial-time solvable linear Diophantine problems

We obtain a polynomial-time algorithm that, given input (A, b), where A=(B|N) is an integer mxn matrix, m<n, with nonsingular mxm submatrix B and b is an m-dimensional integer vector, finds a nonnegative integer solution to the system Ax=b or determines that no such solution exists, provided that b is located sufficiently "deep" in the cone generated by the columns of B. This result improves on some of the previously known conditions that guarantee polynomial-time solvability of linear Diophantine problems.Comment: This is a generalisation and extension of the results from the previous version. At the time the previous version was written, the author was not aware of the results of Brimkov [5