2022/01/03 by Orponen, Tuomas
#11B30 #28A80 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2201.00564
The purpose of this paper is to complete the proof of the following result. Let 0 < β ≤ α < 1 and κ > 0. Then, there exists η > 0 such that whenever A,B ⊂ ℝ are Borel sets with dimH A = α and dimH B = β, then dimH \c ∈ ℝ : dimH (A + cB) ≤ α + η\ ≤ \tfracα - β1 - β + κ. This extends a result of Bourgain from 2010, which contained the case α = β. This paper is a sequel to the author's previous work from 2021 which, roughly speaking, established the same result with dimH (A + cB) replaced by dimB(A + cB), the box dimension of A + cB. It turns out that, at the level of δ-discretised statements, the superficially weaker box dimension result formally implies the Hausdorff dimension result.