An explicit, choice-free construction gives an uncountable proper subring of the real numbers with Hausdorff dimension zero.
Falconer, Fractal geometry, Mathematical foundations and applications(Second Edition)
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An uncountable subring of $\mathbb R$ with Hausdorff dimension zero
An explicit, choice-free construction gives an uncountable proper subring of the real numbers with Hausdorff dimension zero.