On the sum of two affine Cantor sets

TL;DR

This paper investigates the geometric and measure-theoretic properties of sums of affine Cantor sets, revealing their dimension equality, measure finiteness, and typical structural forms under various conditions.

Contribution

It establishes the equality of box and Hausdorff dimensions for sums of affine Cantor sets, characterizes measure properties for almost all pairs, and classifies generic sum structures.

Findings

Sum sets have equal box and Hausdorff dimensions.

For most pairs with sum of dimensions ≤ 1, certain sum sets have zero Hausdorff measure.

Five generic structures are identified for sums of affine Cantor sets with two increasing maps.

Abstract

Suppose that $K$ and $ K'$ are two affine Cantor sets. It is shown that the sum set $K+K'$ has equal box and Hausdorff dimensions and in this number named $s$, $H^s(K+K')<\infty$. Moreover, for almost every pair $(K,K')$ satisfying $HD(K)+HD(K')\leq 1$, there is a dense subset $D\subset \mathbb R$ such that $H^s(K+\lambda K')=0$, for all $\lambda\in D$. It also is shown that in the context of affine Cantor sets with two increasing maps, there are generically (topological and almost everywhere) five possible structures for their sum: a Cantor set, an L, R, M-Cantorval or a finite union of closed intervals.