*N*-fold sums of Cantor sets

## Kathryn E Hare and Toby C O'Neil

## 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.

### Comments

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.

Back to papers | Back to the top

If you have any queries about my work or would like a copy of this paper then please contact me at:

t.c.oneil@open.ac.uk
**Created:***15 August 2000*

**Modified:***9 November 2012*