Discrepancy of Cartesian Products of Arithmetic Progressions

We determine the combinatorial discrepancy of the hypergraph H of cartesian products of d arithmetic progressions in the [N ]d–lattice ([N ] = {0, 1, . . . ,N − 1}). The study of such higher dimensional arithmetic progressions is motivated by a multi-dimensional version of van der Waerden’s theorem, namely the Gallai-theorem (1933). We solve the discrepancy problem for d–dimensional arithmetic progressions by proving disc(H) = Θ(N d4 ) for every fixed integer d ≥ 1. This extends the famous lower bound of Ω(N1/4) of Roth (1964) and the matching upper bound O(N1/4) of Matoušek and Spencer (1996) from d = 1 to arbitrary, fixed d. To establish the lower bound we use harmonic analysis on locally compact abelian groups. For the upper bound a product coloring arising from the theorem of Matoušek and Spencer is sufficient. We also regard some special cases, e.g., symmetric arithmetic progressions and infinite arithmetic progressions. ∗Mathematisches Seminar II; Christian-Albrechts-Universität zu Kiel; Christian-Albrechts-Platz 4; 24098 Kiel; Germany; e-mail: {bed,asr}@numerik.uni-kiel.de †Supported by the graduate school ‘Effiziente Algorithmen und Multiskalenmethoden’, Deutsche Forschungsgemeinschaft ¶SAP AG Düsseldorf, e-mail: [email protected] the electronic journal of combinatorics 11 (2004), #R5 1