6 research outputs found
Directional discrepancy in two dimensions
In the present paper, we study the geometric discrepancy with respect to
families of rotated rectangles. The well-known extremal cases are the
axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in
all possible directions (polynomial discrepancy). We study several intermediate
situations: lacunary sequences of directions, lacunary sets of finite order,
and sets with small Minkowski dimension. In each of these cases, extensions of
a lemma due to Davenport allow us to construct appropriate rotations of the
integer lattice which yield small discrepancy
Low-discrepancy sequences for piecewise smooth functions on the two-dimensional torus
We produce explicit low-discrepancy infinite sequences which can be used to
approximate the integral of a smooth periodic function restricted to a convex
domain with positive curvature in R^2. The proof depends on simultaneous
diophantine approximation and a general version of the Erdos-Turan inequality.Comment: 14 pages, 2 figure
Directional discrepancy in two dimensions
In the present paper, we study the geometric discrepancy with respect to families of rotated rectangles. The well-known extremal cases are the axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in all possible directions (polynomial discrepancy). We study several intermediate situations: lacunary sequences of directions, lacunary sets of finite order, and sets with small Minkowski dimension. In each of these cases, extensions of a lemma due to Davenport allow us to construct appropriate rotations of the integer lattice which yield small discrepancy
Discrepancy with respect to convex polygons
We study the problem of discrepancy of finite point sets in the unit square with respect to convex polygons, when the directions of the edges are fixed, when the number of edges is bounded, as well as when no such restrictions are imposed. In all three cases, we obtain estimates for the supremum norm that are very close to best possible.11 page(s