I have a set of the following form $N = \{ (x_1, ...., x_n, f_1(x), ..., f_s(x): x \in (-1, 1)^n \}$ with each $f_j$ smooth. This is an $n$ dimensional manifold. I think it is the case that the $d$ dimension Hausdorff measure of $N$ is infinity as soon as $d < n$, that is to say $H^d(N) = \infty$.
I think it just amounts to finding a suitable cover of $N$, but I have been struggling to prove it.. any pointers or reference would be appreciated!