2018/10/26 by Hambrook, Kyle, Taylor, Krystal
#28A75 #28A80 #42B10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1810.11553
We investigate the Lebesgue measure, Hausdorff dimension, and Fourier dimension of sets of the form RY + Z, where R ⊆ (0,∞) and Y, Z ⊆ ℝd. We prove a theorem on the Lebesgue measure and Hausdorff dimension of RY+Z; The theorem is a generalized variant of some theorems of Wolff and Oberlin in which Y is the unit sphere, but its proof is much simpler. We also prove a deeper existence theorem: For each α∈ [0,1] and for each non-empty compact set R ⊆ (0,∞), there exists a compact set Y ⊆ [1,2] such that dimF(Y) = dimH(Y) = dimM(Y) = α and dimF(RY) ≥ min\ 1, dimF(R) + dimF(Y)\. This theorem verifies a weak form of a more general conjecture, and it can be used to produce new Salem sets from old ones.