Given an arbitrary number $\delta \in (0, 1)$, are there attractors of IFS $F$ with Hausdorff dimension $HD(F) = \delta$?

Question

Given an arbitrary number $\delta \in (0, 1)$, are there fractals $F$ obtained as limit sets of iterated function systems in $\mathbb{R}$ with Hausdorff dimension $HD(F) = \delta$?

My work: Let $\delta \in (0, 1)$. Suppose that there exists an IFS $\{ S_1, ..., S_m \}$ with $F$ as limit set and $HD(F) = \delta$. Suppose that this IFS satisfies the open set condition and the contractions $S_1, ..., S_m$ have ratios $r_i, \; i = 1, ..., m$. Then from Hutchinson theorem we have that $$r_1^\delta + ... r_m^\delta = 1.$$ I think that the answer is NO, but I don't know how to obtain a contradiction. Can someone help me? Thank you!

Answer 1

Try $m=2$ and $r_1=r_2=(1/2)^{1/\delta}$.