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 $1$?
Best Answer
For any $r<1$, you can construct a Cantor set with Hausdorff dimension $r$ by varying the lengths of the intervals in the usual Cantor set construction. In particular, you can let $C_n\subset[0,1]$ be a Cantor set of Hausdorff dimension $1-1/n$ for each $n$. The union $C=\bigcup C_n$ then has Lebesgue measure $0$ because each $C_n$ does, but Hausdorff dimension $1$.