Constructs additive subgroups of reals with prescribed Hausdorff dimension that generate the whole line under scalar addition, proves subrings cannot do so unless they are all of R, and gives CH-based zero-dimensional examples with sharp covering characterizations.
Hausdorff dimension and countable Borel equivalence relations
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When is $A + x A =\mathbb{R}$
Constructs additive subgroups of reals with prescribed Hausdorff dimension that generate the whole line under scalar addition, proves subrings cannot do so unless they are all of R, and gives CH-based zero-dimensional examples with sharp covering characterizations.