N-fold sums of Cantor sets

Kathryn E Hare and Toby C O'Neil


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.

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Created:15 August 2000
Modified:9 November 2012