Geodesic
Structural Szemerédi-Trotter for lattices and their generalizations
Shival Dasu1 ; Adam Sheffer2 ; Junxuan Shen3
1University of Indiana at Bloomington
2City University of New York
3California Institute of Technology
The electronic journal of combinatorics, Tome 32 (2025) no. 1
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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 Szemerédi-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.
DOI : 10.37236/12466
Classification : 11H06, 52C10
Mots-clés : Szemerédi-Trotter problem, arithmetic progression

Shival Dasu  1   ; Adam Sheffer  2   ; Junxuan Shen  3

1 University of Indiana at Bloomington
2 City University of New York
3 California Institute of Technology 
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     title = {Structural {Szemer\'edi-Trotter} for lattices and their generalizations},
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Shival Dasu; Adam Sheffer; Junxuan Shen. Structural Szemerédi-Trotter for lattices and their generalizations. The electronic journal of combinatorics, Tome 32 (2025) no. 1. doi: 10.37236/12466

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