On Sums of Nearly Affine Cantor Sets

TL;DR

This paper proves that the sum of a compact set and a family of nearly affine Cantor sets typically has positive measure when their dimensions sum to more than one, extending results about affine Cantor sets.

Contribution

It establishes that sums of nearly affine Cantor sets with dimensions summing to more than one have positive Lebesgue measure under broad conditions.

Findings

Sum of a compact set and nearly affine Cantor sets has positive measure when dimensions sum > 1.

Generically, sum of two affine Cantor sets has positive Lebesgue measure if their dimensions sum > 1.

Results hold under natural technical conditions for dynamically defined Cantor sets.

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 $\text{dim}_H\, K+\text{dim}_H\, C_\lambda>1$ 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 one.