Structural Szemerédi-Trotter for Lattices and their Generalizations

• DOI: 10.37236/12466
• Publication Date: 2025-03-16T21:22:26Z
• Authors: Shival Dasu; Adam Sheffer; Junxuan Shen

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