Is it possible to construct a smooth curve with fractional Hausdorff dimension?

Question

It is known that fractal curves have fractional Hausdorff dimension. These curves are not smooth and have undefined length. However, is the converse true?

If a curve has a fractional Hausdorff dimension, then must it be non-smooth/not differentiable?

Answer 1

A rectifiable curve has $\sigma$-finite and nonzero $1$-dimensional Hausdorff measure (essentially, the length is what $1$-dimensional Hausdorff measure is). Therefore it must have Hausdorff dimension $1$. Thus a curve whose Hausdorff dimension is not $1$ can't be rectifiable, and certainly can't be smooth.

Answer 2

If $f$ is continuous then you can cover $f$ with boxes of height $2\epsilon$ (and varying width) around points of the graph. For compact domain, a finite and hardly ovelapping subcover suffices, showing the graph has Hausdorff measure $0$.