ca.analysis and odes - Measure 0 sets on the line with Hausdorff dimension 1

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.