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arXiv · 1812.09547

Sum-Product Phenomena for Planar Hypercomplex Numbers

Matthew Hase-Liu·Adam Sheffer
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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 Szemeredi-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.

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Matthew Hase-Liu, Adam Sheffer. 2018-12-22. Sum-Product Phenomena for Planar Hypercomplex Numbers. https://arxiv.org/abs/1812.09547

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