An explicit, choice-free construction gives an uncountable proper subring of the real numbers with Hausdorff dimension zero.
Falconer, On the Hausdorff dimensions of distance sets, Mathematika 32 (1985), 206–212
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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.