Cantor sets with Hausdorff dimension 1 or 0 скачать в хорошем качестве

Cantor sets with Hausdorff dimension 1 or 0

In previous video, for every number alpha between 0 and 1 we constructed a Cantor set with Hausdorff dimension equal to alpha. But do there exist Cantor sets with Hausdorff dimension equal to 1 or to zero? Here we show how adapting the ratios of the removed sub-intervals we cab achieve this. We use generalized Hausdorff measures to catch the precise size of these sets. Case of dim=1 is interesting as we have a full dimensional subset that is nowhere dense (but see follow-up video for more surprise with fat Cantor sets). Case of dim=0 is interesting because it gives ab example of ab uncountable set with Hasudorff dim zero -- countable and finite sets are obviously of dim zero.