2017/07/05 by Károly Simon, Simon, Károly, Krystal Taylor +1 · 1 citation
Mathematics · #28a75 (primary) #28a80 #28a99 (secondary) #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1707.01407
openalex publication_date 2017/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the measure and dimension of A+Γ:=\a+v:a∈ A, v∈ Γ\ when A⊂ ℝ2 and Γ is a piecewise C2 curve. Assuming Γ has non-vanishing curvature, we verify that (a) if dim\rm H A ≤ 1, then dim\rm H (A+Γ)=dim\rm H A +1, where dim\rm H denotes the Hausdorff dimension; (b) if dim\rm H A>1, then Leb2(A+Γ)>0, where Leb2 denotes the 2-dimensional Lebesgue measure; (c) if dim\rm H A=1 and H1(A) < ∞, then Leb2(A+Γ)=0 if and only if A is an irregular (purely unrectifiable) 1-set. Here, H1 denotes the 1-dimensional Hausdorff measure. Items (a) and (b) follow from previous works of Wolff and Oberlin using Fourier analysis. In this article, we develop an approach using nonlinear projection theory which gives new proofs of (a) and (b) and the first proof of (c). Item (c) has a number of consequences: if a circle is thrown randomly on the plane, it will almost surely not intersect the four corner Cantor set. Moreover, the pinned distance set of an irregular 1-set has 1-dimensional Lebesgue measure equal to zero at almost every pin t∈ ℝ2.