I will use $dimA$ to denote the Hausdorff dimension of a set $A \subseteq \mathbb{R}$. Being a null set means having Lebesgue measure zero.

In the 1966 paper "Additive gruppen mit vorgegebener hausdorffscher dimension" by Paul Erdös and Bodo Volkmann, it was shown that under the continuum hypothesis $(2^{\aleph_0} = \aleph_1)$ there is an additive subgroup $G$ of $\mathbb{R}$ such that

(i) $G$ is non-null. In particular, $dimG = 1$.

(ii) If $A$ is a null set, then $G \cap A$ is countable. In particular, there does not exist a subgroup $H$ of $G$ with $dimH \in (0,1)$.

My question is as follows: Assuming only ZFC, is there an additive subgroup $G$ of $\mathbb{R}$ such that (i) dim $G$ = 1 and (ii) there does not exist a subgroup $H$ of $G$ with $dimH \in (0,1)$.

The proof of Erdös and Volkmann starts by enumerating the collection of null $G_\delta$ sets, and their argument relies heavily on the continuum hypothesis. However, their result appears to be stronger than the question I am asking. The 1984 paper "A Very Sparse Set of Dimension 1" by Anatole Beck seems to be a step in the right direction, but I've found little information otherwise.


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