Here's an interesting problem one can formulate for a student. This problem arises when considering special ergodic theorems:

*On a finite dimensional manifold $M$ with a Lebesgue measure $\mu$, does every measure zero set equal a countable union of the sets of less than full Hausdorff dimension*?

For a diffeomorphism $f$ of $M$ and a continuous function $\varphi$ on $M$, define $$\overline \varphi = \lim_{n \rightarrow \infty} \frac{1}{n} \sum_0^{n-1} \varphi \circ f^k(x).$$ Then the Birkhoff theorem asserts that for almost all $x$, $\overline \varphi \rightarrow \int_M \varphi, n \rightarrow \infty$. But consider the set $K_{\alpha}$ of $x$ where $$\alpha \leq |\overline \varphi - \int \varphi|.$$ So Birkhoff says $\mu(K_\alpha)=0$, but what about the Hausdorff dimension of $\mu(K_\alpha)=0$? For some diffeomorphisms, for example hyperbolic maps, it was proven that $\dim_H K_\alpha < \dim X$. That fact gives a rise to my question. I expect a negative answer, but I can not find a counterexample.