A New Upper Bound for the Heilbronn Triangle Problem
summary
Proves that for sufficiently large n every n-point set in the unit square contains a triangle of area < n^{-8/7-1/2000} by combining an incidence-based high-low method with a stronger initial incidence estimate coming from discretized projection theory. Key elements are partitioning into u-by-u cells, counting incidences between points and narrow tubes (width ≈ 4δ/length), using Rees energy sums and a weighted incidence graph, and importing the Orponen‑Shmerkin‑Wang discretized Marstrand/Bourgain sum‑product input to force many well spaced directions.
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chapters (10)
- 0:00Introduction to the problem
- 2:16Erdős and early constructions
- 5:46Geometric upper bounds
- 11:12The role of incidence geometry
- 21:19New polynomial improvements
- 23:42Sum-product and projections
- 30:26Inductive and initial steps
- 37:23High-low method mechanics
- 43:25Discrete Marstrand theorem
- 57:46Conclusion and open questions
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