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- W1704524108 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." @default.
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- W1704524108 date "2004-01-02" @default.
- W1704524108 modified "2023-10-14" @default.
- W1704524108 title "Discrepancy of Cartesian Products of Arithmetic Progressions" @default.
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- W1704524108 doi "https://doi.org/10.37236/1758" @default.
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