Seminar

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Hausdorff measure $\mathcal{H}^\alpha (E)$ of a set $E$ is a nonnegative number that measures its ``$\alpha$-dimensional'' size. If $E$ is a curve, $\mathcal{H}^1 (E)$ is the \textbf{length} of the curve, as we know it from Calc 2. But $\mathcal{H}^2 (E) = 0$, meaning our 2-dimensional goggles will not notice this set. Hausdorff measure applies to any arbitrary set $E$ and more surprisingly, $\alpha$ does not have to be an integer. Thus you may measure $\sqrt{2}$-, $\frac{2}{3}$-, or even $\pi$-dimensional size of a set.

Take your favorite positive integer. If it's even, divide it by 2. If it's odd, multiply by 3 and add 1. Continue these steps with the new number you get and repeat forever. The Collatz conjecture claims that any positive integer you start with, you will eventually end up at 1. Recently, Terence Tao proved some strong partial results on this conjecture. In light of this, I will remark on some (not new) theorems involving this infamous conjecture.

Part 2 of Lectures of Professor Manuel Ritoré from the University of Granada, Spain

Lectures of Professor Manuel Ritoré from the University of Granada, Spain