A Proto Inverse Szemerédi–Trotter Theorem

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• Olivine Silier (UC Berkeley)
• Wednesday 06 November 2024, 13:30-15:00
• MR4, CMS.

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A point-line incidence is a point-line pair such that the point is on the line. The Szemerédi-Trotter theorem says the number of point-line incidences for n (distinct) points and lines in R² is tightly upperbounded by O(n^4/3). We advance the inverse problem: we geometrically characterize ‘sharp’ examples which saturate the bound by proving the existence of a nice cell decomposition we call the two bush cell decomposition. The proof crucially relies on the crossing number inequality from graph theory and has a traditional analysis flavor.

Our two bush cell decomposition also holds in the analogous point-unit circle incidence problem. This constitutes an important step towards obtaining an ε improvement in the unit-distance problem.

This talk is part of the Discrete Analysis Seminar series.

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