I use $\dim\_H(E)$ to denote the Hausdorff dimension of a set $E \subseteq \mathbb{R}$ and $|E|$ to denote its Lebesgue measure. It is easy to see from the definition of Hausdorff dimension that if $\dim\_H(E) < 1$, then $|E| = 0$. The converse is not true, and there are many cases where $\dim\_H(E) = 1$ yet $|E| = 0$. So the question:


**What was the first (or most elementary) example of this phenomenon?**


After some looking around, I was able to prove that a central Cantor set $C$ with ratio of dissection $r\_k = 1/(2+\frac{1}{k})$ satisfies the condition I want. It is easy to see that $|C| = 0$ since at step $n$ of the process to construct this Cantor set, it has measure $2^n(r\_1 \cdots r\_n)$ which in this case limits to 0, but for the Hausdorff dimension I required a non-trivial result from the paper *Sums of Cantor sets* (Cabrelli, Hare, Molter) that gave the formula 


$\dim\_H(C) = \liminf\_n \frac{n \ln 2}{\ln r\_1 \cdots r\_n}$.


This result is fairly recent and sophisticated, and I feel that there should be older and simpler examples.



Measure 0 sets on the line with Hausdorff dimension 1
Try a countable union of sets (such as Cantor sets) whose Hausdorff dimension tends to 1. 


