44 research outputs found
Cyclic shifts of the van der Corput set
In [13], K. Roth showed that the expected value of the discrepancy of
the cyclic shifts of the point van der Corput set is bounded by a constant
multiple of , thus guaranteeing the existence of a shift with
asymptotically minimal discrepancy, [11]. In the present paper, we
construct a specific example of such a shift.Comment: 12 page
Estimates of the Discrepancy Function in Exponential Orlicz Spaces
We prove that in all dimensions n at least 3, for every integer N there
exists a distribution of points of cardinality , for which the associated
discrepancy function D_N satisfies the estimate an estimate, of sharp growth
rate in N, in the exponential Orlicz class exp)L^{2/(n+1)}. This has recently
been proved by M.~Skriganov, using random digit shifts of binary digital nets,
building upon the remarkable examples of W.L.~Chen and M.~Skriganov. Our
approach, developed independently, complements that of Skriganov.Comment: 13 page
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