Discrepancy of Cartesian Products of Arithmetic Progressions

Abstract

We determine the combinatorial discrepancy of the hypergraph ${\cal H}$ of cartesian products of $d$ arithmetic progressions in the $[N]^d$–lattice ($[N] = \{0,1,\ldots,N-1\}$). 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 ${\rm disc}({\cal H}) = \Theta(N^{d/4})$ for every fixed integer $d \ge 1$. This extends the famous lower bound of $\Omega(N^{1/4})$ of Roth (1964) and the matching upper bound $O(N^{1/4})$ 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.