Sum–product phenomena for planar hypercomplex numbers

Abstract

We study the sum–product problem for the planar hypercomplex numbers: the dual numbers and double numbers. These number systems are similar to the complex numbers, but it turns out that they have a very different combinatorial behavior. We identify parameters that control the behavior of these problems, and derive sum–product bounds that depend on these parameters. For the dual numbers we expose a range where the minimum value of max{|A+A|,|AA|} is neither close to |A| nor to |A|2.To obtain our main sum–product bound, we extend Elekes’ sum–product technique that relies on point–line incidences. Our extension is significantly more involved than the original proof, and in some sense runs the original technique a few times in a bootstrapping manner. We also study point–line incidences in the dual plane and in the double plane, developing analogs of the Szemerédi–Trotter theorem. As in the case of the sum–product problem, it turns out that the dual and double variants behave differently than the complex and real ones.