I would like to ask the following:
Are there "many" sets, say in the interval $[0,1]$, with zero Lebesgue measure but with Hausdorff dimension $1$?
The motivation for this question is the dichotomy between measure and category. There are certainly dense sets with zero Lebesgue measure. But a dense set need not have positive Hausdorff dimension (for example, the rationals are dense but have zero Hausdorff dimension).
Honestly, I would already be satisfied with an answer to the following question:
Is there any set in $[0,1]$ with zero Lebesgue measure but with Hausdorff dimension $1$?