Structural Szemer\'edi-Trotter for Lattices and their Generalizations
Combinatorics
2023-10-03 v1
Abstract
We completely characterize point--line configurations with 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.
Keywords
Cite
@article{arxiv.2310.00191,
title = {Structural Szemer\'edi-Trotter for Lattices and their Generalizations},
author = {Shival Dasu and Adam Sheffer and Junxuan Shen},
journal= {arXiv preprint arXiv:2310.00191},
year = {2023}
}