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    $\begingroup$ Perhaps I must explain that the existence of $A\subset{\bf R}$ with $0<H_f(A)<1$ is well known and due to Dvoretzky: Dvoretzky, A. A note on Hausdorff dimension functions. Proc. Cambridge Philos. Soc. 44, (1948). 13–16. $\endgroup$ Commented Mar 29, 2012 at 19:49
  • $\begingroup$ Thank you so much for your answer, as far as I understand, for $\mathbb{R}^n$ we have to consider $f_n(x)=x^n \log (e/x)$ and everything will work as it has to. And the number e in the definition of f doesn't value much - we can take any positive number we want. $\endgroup$ Commented Apr 3, 2012 at 19:54
  • $\begingroup$ Yes, the constant $e$ is only to make $x \log(e/x)$ monotone on $[0,1]$. In fact $H_f = H_g$ if $f$ and $g$ coincide on an interval $[0,\varepsilon]$, therefore there is some liberty in choosing $f$. $\endgroup$ Commented Apr 5, 2012 at 11:09