Antoine's Necklace and positive Hausdorff/Lebesgue measure

Question

I have the following question:

The usual construction of the Antoine's Necklace produces a Cantor of $1$-dimensional Hausdorff measure in $\mathbb R^3$.

I would like to know whether one could adapt the construction to produce a larger Cantor set, namely a antoine's necklace (or similar constructions) with positive $3$-dimensional Hausdorff measure or Lebesgue measure.

This looks very plausible for me since we have freedom to determine the sizes of the linked chain torus at each step and topologically the construction can be proceeded as in the usual Cantor cube constructions.

If this is already known, precise references are greatly appreciated. Comments and suggestions are also warmly welcome. Thanks.

Answer 1

It can be done for the very same reason you mention: given a torus, one can find inside it four linked tori of arbitrarily large relative measure. So one can proceed as in the standard construction of a Cantor set of positive Lebesgue measure.

More explicitly, fix a sequence $r_n\in (0,1)$ such that $\prod_{n=1}^\infty r_n=1/2$ (say). Start with a torus $T_0\subset\mathbb{R}^3$ of unit volume. Replace $T_0$ by the union $T_1$ of four linked tori $T_{1,1},\ldots, T_{1,4}$ whose union has measure $r_1$. Then replace each $T_{1,i}$ by four linked tori $T_{1,i,j}$ contained in $T_{1,i}$ whose union has measure $r_1 r_2$; let $T_2$ be the union of these 16 tori. If you continue inductively, the intersection $\cap_n T_n$ is an Antoine necklace of measure $1/2$.

Edit: I'm not so sure this actually works. One needs to be a little careful to ensure that the diameters of the tori that make up $T_n$ go to zero as $n\to\infty$ (which is needed to ensure the resulting set is totally disconnected).