Lattice points in unbounded point sets

Counting Lattice Points in Admissible Adelic Sets

Representations of lattice point sets

Economical coverings of sets of lattice points

Let t(n, d) be the minimum number t such that there are t of the n lattice points {(x1, . . . , xd) : 1 ≤ xi ≤ n} so that the ( t 2 ) lines that they determine cover all the above n lattice points. We prove that for every integer d ≥ 2 there are two positive constants c1 = c1(d) and c2 = c2(d) such that for every n c1n d(d−1)/(2d−1) ≤ t(n, d) ≤ c2n log n. The special case d = 2 settles a proble...

Bisections of Two Sets of Points in the Plane Lattice

Assume that 2m red points and 2n blue points are given on the lattice Z2 in the plane R2. We show that if they are in general position, that is, if at most one point lies on each vertical line and horizontal line, then there exists a rectangular cut that bisects both red points and blue points. Moreover, if they are not in general position, namely if some vertical and horizontal lines may conta...

Lattice Points in Large Borel Sets and Successive Minima

Let B be a Borel set in E with volume V (B) = ∞. It is shown that almost all lattices L in E 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 E with V (S) = ∞. Then for almost all lattices L in E the successive minima λ1(S,L), . . . , λd(S,L) of S with respect to L are 0...