On sums of nearly affine Cantor sets
Volume 240 / 2018
Fundamenta Mathematicae 240 (2018), 205-219
MSC: Primary 28A80, 37D99, 28A78.
DOI: 10.4064/fm183-3-2017
Published online: 7 August 2017
Abstract
For a compact set $K\subset \mathbb {R}^1$ and a family $\{C_\lambda \}_{\lambda \in J}$ of dynamically defined Cantor sets sufficiently close to affine with $\operatorname {dim}_{\rm H} K +\operatorname {dim}_{\rm H} C_\lambda \gt 1$ for all $\lambda \in J$, under natural technical conditions we prove that the sum $K+C_\lambda $ has positive Lebesgue measure for almost all values of the parameter $\lambda $. 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 $1$.
