Discrepancy of arithmetic structures
In discrepancy theory, we investigate how well a desired aim can be achieved. So typically we do not compare our solution with an optimal solution, but rather with an (idealized) aim. For example, in the declustering problem, we try to distribute data on parallel disks in such a way that all of a prespecified set of requests find their data evenly distributed on the disks. Hence our (idealized) aim is that each request asks for the same amount of data from each disk. Structural results tell us to which extent this is possible. They determine the discrepancy, the deviation of an optimal solution from our aim. Algorithmic results provide good declustering scheme. We show that for grid structure data and rectangle queries, a discrepancy of order (log M)^((d-1)/2) cannot be avoided. Moreover, we present a declustering scheme with a discrepancy of order (log M)^(d-1). Furthermore, we present discrepancy results for hypergraphs related to the hypergraph of arithmetic progressions, for the hypergraph of linear hyperplanes in finite vector spaces and for products of hypergraphs.