As usual, for a function $f: \mathbb{R} \to \mathbb{R}$ the graph of $f$ is the set $\{ (x,f(x)) \mid x \in \mathbb{R} \}$. For which continuous nowhere differentiable functions $f: \mathbb{R} \to \mathbb{R}$ does the graph of $f$ have fractional fractal (Box or Hausdorff) dimension?

The Weirstrass function has fractional Hausdorff dimension. But, as explained in this answer, one can construct examples of such functions with integer fractal dimension.

Are there sufficient conditions for such functions to have fractional fractal dimension?

  • $\begingroup$ Related: the box counting dimension of $\sin\frac{1}{x}$ is equal to $3/2$. $\endgroup$
    – corey979
    Commented Jul 7, 2018 at 21:31
  • $\begingroup$ Being "uniform wiggly" is enough: $\exists c>0$ and $p\in (0, 1)$ such that $\operatorname{osc}_{[a, b]}f \ge c(b-a)^p$ for all intervals $[a, b]$. But I'm not about to write a proof of that. (Also, this is sufficient to have dimension greater than 1; it may be equal to $2$.) $\endgroup$
    – user357151
    Commented Jul 7, 2018 at 22:18


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