N-fold sums of Cantor sets

Abstract

We generalize the Newhouse gap lemma by finding a geometric condition which ensures that $$N$$-fold sums of compact sets, which might even have thickness zero, are intervals. We also obtain a new proof of a lower bound on the thickness of the sum of two Cantor sets.

For a compact set $$C\subseteq \mathbb{R}$$, a gap of $$C$$ is a non-empty connected component of the complement of $$C$$. A bridge of $$C$$ is a non-empty, closed interval whose left and right endpoints are left-isolated and right-isolated, respectively, in $$C$$. A bridge and gap which share a boundary point are called a bridge-gap pair.

A bridge-gap pair $$(B,G)$$ is maximal if the bridge $$B$$ is a maximal interval which contains no point of a gap whose length is at least the length of $$G$$. We let $$g(B)$$ be the length (possibly infinite) of the smallest component of the complement of $$C$$ adjacent to the bridge $$B$$. Notice that when $$(B,G)$$ is a maximal bridge-gap pair, then $$g(B)=\left| G\right|$$ (where $$\left| \cdot \right|$$ denotes the diameter).

For $$N\geq 1$$ we say that non-empty compact sets $$C_{1},...,C_{N}$$ satisfy the $$N$$-fold bridge-gap condition if whenever $$(B_{i},G_{i})$$ are maximal bridge-gap pairs of $$C_{i}$$ for $$i=1,\ldots,N$$ with $$\min\{|G_i|:1\leq i\leq N\}<\infty$$, then $\min_{k}\{\left| B_{k}\right| +\left| G_{k}\right| \}\leq \sum_{1\leq i\leq N}\left| B_{i}\right| \text{.}$

The main result in the paper is:

Suppose that $$C_1,\ldots, C_N$$ are non-empty compact sets for which the $$N$$-fold bridge-gap condition holds. Then $C_1+\cdots +C_N =\{c_1+\cdots +c_N :c_i\in C_i,\, i=1,\ldots, N\}$ is an interval.