On Sums of Nearly Affine Cantor Sets

2015/10/23 by Anton Gorodetski, Gorodetski, Anton, Scott Northrup +1
Mathematics · #28A78 #28A80 #37D99 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #math.CA #math.DS #msc:28A78 #msc:28A80 #msc:37D99

paper · pdf · doi:10.48550/arxiv.1510.07008

15 pages, 1 figure

arxiv created 2015/10/23 · arxiv updated 2015/10/26

Abstract

For a compact set K⊂ ℝ¹ and a family \C_λ\_{λ∈ J} of dynamically defined Cantor sets sufficiently close to affine with dim_H K+dim_H C_λ>1 for all λ∈ J, under natural technical conditions we prove that the sum K+C_λ has positive Lebesgue measure for almost all values of the parameter λ. As a corollary, we show that generically the sum of two affine Cantor sets has positive Lebesgue measure provided the sum of their Hausdorff dimensions is greater than one.

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