*[Corrected, thanks – T.]*

I am having problem with exercise 5, especifically for the unbounded generalization of the definion of Minkowski dimension. Any hint? Thank you very much.

]]>The point is that if is a sequence of nonnegative numbers such that is large, then one must have for some (pigeonhole principle). In this question, the numbers are basically the Frostman measure of the union of balls of radius equal (after rounding) to in a suitable covering of the set by balls of variable radius.

]]>I’m struggling with Exercise 21.(i). I don’t see how being small is used. I see directly that the lower Minkowski dimension of any set with positive measure is greater or equal to , but I guess the problem is treating covering by balls of varying radius.

]]>In Exercise 20, the statement asserted to be equivalent to seems independent to the set . It may be better to correct this statement.

*[Corrected, thanks – T.]*

Use Trick 11 (combined with Trick 2).

]]>Looks like Trick 6 in

https://terrytao.wordpress.com/2010/10/21/245a-problem-solving-strategies/

One can show this by covering the bounded subset of with balls of a smaller radius. It then follows by the observation in the previous comment that

for all bounded subsets. How to “patch things together”?

Without countably subadditive, how can one pass it to the unbounded case? (Stein-Shakarchi established subadditivity; one could look it up in the book. Can one do it alternatively?) Or does one have to establish some “continuity” of (which I don’t find a way to show) so that one can look at the Hausdorff outer measure of sets ?

(It’s interesting to look at different proofs. In Wolff’s notes that mentioned in the last sentence of this post, the author gave a mysterious one-sentence proof: “when , one can cover by discs with arbitrarily small.”, which I have no ideas what he means.)

]]>First show that any bounded subset of has finite -dimensional Hausdorff outer measure.

]]>*[Corrected, thanks – T.]*

This shows that