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We construct a compact set C of Hausdorff dimension zero such that cof(𝒩) many translates of C cover the real line. Hence it is consistent with ZFC that less than continuum many translates of a zero-dimensional compact set can cover the real line. This answers a question of Dan Mauldin.

A hull of A ⊆ [0,1] is a set H containing A such that λ*(H) = λ*(A). We investigate all four versions of the following problem. Does there exist a monotone (with respect to inclusion) map that assigns a Borel/$G_{\delta }$ hull to every negligible/measurable subset of [0,1]? Three versions turn out to be independent of ZFC, while in the fourth case we only prove that the nonexistence of a monotone $G_{\delta }$ hull operation for all measurable sets is consistent. It remains open whether existence here is also consistent....

Suppose $F\subset [0,1]$ is closed. Is it true that the typical (in the sense of Baire category) function in $C^1[0,1]$ is one-to-one on $F$? If ${\underline{dim}}_{B}F<1/2$ we show that the answer to this question is yes, though we construct an $F$ with ${dim}_{B}F=1/2$ for which the answer is no. If $C_{\alpha }$ is the middle-$\alpha$ Cantor set we prove that the answer is yes if and only if $dim\left(C_{\alpha }\right)\le 1/2.$ There are $F$’s with Hausdorff dimension one for which the answer is still yes. Some other related results are also presented.