A new bound on partial sum-sets and difference-sets, and applications to the Kakeya conjecture

1999/06/14 by Nets Hawk Katz, Terence Tao, Katz, Nets Hawk +1
Mathematics · #05C35 #42B25 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #math.CA #math.CO #msc:05C35 #msc:42B25

paper · pdf · doi:10.48550/arxiv.math/9906097

6 pages, submitted to Math Research Letters; improved bounds in revised version; typoes corrected in second revised version

openalex publication_date 1999/06/14 · arxiv created 2000/01/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A, B, be finite subsets of an abelian group, and let G ⊂ A × B be such that # A, # B, # \a+b: (a,b) ∈ G \ ≤ N. We consider the question of estimating the quantity # \a-b: (a,b) ∈ G \. Recently Bourgain improved the trivial upper bound of N² to N^2-1/13, and applied this to the Kakeya conjecture. We improve Bourgain's estimate further to N^2-1/6, and obtain the further improvement of N^2-1/4 if we also know that # \a+2b: (a,b) ∈ G\ ≤ N. We conclude that Besicovitch sets in \Rⁿ have Hausdorff dimension at least 6n/11+5/11 and Minkowski dimension at least 4n/7 + 3/7. This is new for n > 8.

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