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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?

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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.

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