Does a fractal exist for any given dimension?

Question

For example a Sierpinski Triangle has a dimension of $log_23 \approx 1.58 $, so if I named a dimension such as $log_2301$ is there definitely a fractal that exists in that dimension?

Answer 1

From

Mohsen Soltanifar, On A Sequence of Cantor Fractals, Rose Hulman Undergraduate Mathematics Journal, Vol 7, No 1, paper 9, 2006

Theorem: For any given $r > 0$, there are uncountable fractals with Hausdorff dimension $r$ in $n$-dimensional Euclidean space $\mathbb{R}^n$ ($n \geq - \lfloor -r \rfloor$).

The parenthetical condition just means we don't try to cram a $>n$ dimensional fractal into a $n$ dimensional space.

Answer 2

Yes, there are indeed aleph-two of them !

Soltanifar M. A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals. Mathematics. 2021; 9(13):1546. https://doi.org/10.3390/math9131546

Theorem(The Generalized Hausdorff Dimension Theorem). For any real $r>0$ and $l≥0,$ there are aleph-two (symmetric) fractals with the Hausdorff dimension $r.1_{\{0\}}(l)+n.1_{(0,∞)}(l)$ and Lebesgue measure l in $R^n$ where $(⌈r⌉≤n).$