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arXiv · 2310.00191

Structural Szemer\'edi-Trotter for Lattices and their Generalizations

Shival Dasu·Adam Sheffer·Junxuan Shen
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Abstract

We completely characterize point--line configurations with $\Theta(n^{4/3})$ incidences when the point set is a section of the integer lattice. This can be seen as the main special case of the structural Szemer\'edi-Trotter problem. We also derive a partial characterization for several generalizations: (i) We rule out the concurrent lines case when the point set is a Cartesian product of an arithmetic progression and an arbitrary set. (ii) We study the case of a Cartesian product where one or both sets are generalized arithmetic progression. Our proofs rely on deriving properties of multiplicative energies.

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Shival Dasu, Adam Sheffer, Junxuan Shen. 2023-09-29. Structural Szemer\'edi-Trotter for Lattices and their Generalizations. https://arxiv.org/abs/2310.00191

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