On the discrepancy for cartesian products

Let B-2 denote the family of all circular discs in the plane. It is proved that the discrepancy for the family {B-1 x B-2: B-1, B-2 is an element of B-2} in R-4 is O(n(1/4+epsilon)) for an arbitrarily small constant epsilon > 0, that is, it is essentially the same as that for the family B-2 itself. The result is established for the combinatorial discrepancy, and consequently it holds for the discrepancy with respect to the Lebesgue measure as well. This answers a question of Beck and Chen. More generally, we prove an upper bound for the discrepancy for a family {Pi(i-1)(k) A(i):A(i) is an element of A(i), i = 1, 2,...,k}, where each A(i) is a family in R-di, each of whose sets is described by a bounded number of polynomial inequalities of bounded degree. The resulting discrepancy bound is determined by the 'worst' of the families A(i) and it depends on the existence of certain decompositions into constant-complexity cells for arrangements of surfaces bounding the sets of A(i). The proof uses Beck's partial coloring method and decomposition techniques developed for the range-searching problem in computational geometry.