Fractal Geometry and Hausdorff dimension at first glance appear esoteric and wholly unrelated to other fields of mathematics. However, Hausdorff dimension turns out to be a very powerful tool in distinguishin sets from one another when other more intuitive methods like that of topological properties and measure fail. One such example from Number Theory is that of the distribution of digits in m-expansions on the unit interval. In such an analysis we see that these distribution of digits in m-expansion sets are very similar with respect to topology and Lebesgue measure, but with the help of probability theory and a bit of fractal geometry we'll give a sketch of why these sets are different. Further this sort of analysis can be thought of as a type of "multifractal".

00:00 The Cantor Set: Construction and interpretation in terms of distribution of digits in ternary expansion
02:50 Distribution of Digits in m-expansion sets definition
04:14 The equidistributed digit sets: Normal numbers, an open question, and what they say about the non-equidistributed digit sets
08:00 Using Hausdorff Dimension to Analyze Distribution of Digits in m-expansion sets.
12:00 Multifractal Analysis?

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