Comparing the uniformity invariants of null sets for different measures (Q1771327)

Given any real number \(r\in (0,1)\), the authors construct a model of set theory in which every set of reals of size \(\aleph_1\) is Lebesgue measurable and there is a set of reals of size \(\aleph_1\) which is not a null set with respect to \(r\)-dimensional Hausorff measure. (This answers Question FQ from David Fremlin's list of open questions.) These considerations are motivated by the following nice geometric question posed by P. Komjáth: Suppose that every set of size \(\aleph_1\) has Lebesgue measure zero. Does it follow that the union of any set of \(\aleph_1\) lines in the plane has Lebesgue measure zero? This problem remains open.