How to calculate Assouad Dimension of sets

Question

I am trying to work with sets $F$ as subsets of $\mathbb{R}^n$ and their Assouad dimension. I am a bit stuck on how to apply the definition as it asks that

1. We find the infimum of possible $\alpha$
2. We show for all $x \in F$ there exists a constant $C>$ such that for all $0<r<R$ the following holds

$$N_r(B(x,R)\cap F) \leq C\left(\frac{R}{r}\right)^\alpha$$

This is the definition I am using, which is from Fraser's Assouad Dimension and Fractal Geometry.

My issue is more related to HOW I can show these things. Of course I can let $x \in F$ and $0<r<R$, but then how do I show a constant that I choose will satisfy the inequality, moreover, that the $\alpha$ I choose is the infimum of possibilities.

The only example that Fraser provides does not do a direct proof but uses some sort of contradiction argument, which I am still not sure how?

Any advice or examples would be appreciated. Thank you!

Answer 1

I think your (1) and (2) are in the wrong order.

For a fixed $\alpha$, the claim (2) might be true or false. The Assouad dimension is the infimum of all $\alpha$ such that (2) is true.

This definition allows you to bound the Assouad dimension in the following ways (which may be useful when computing the exact value of the dimension is difficult).

• If are able to show (2) holds for some fixed $\alpha$, then the Assouad dimension is $\le \alpha$.
• If you are able to show (2) is false for some fixed $\alpha$, then the Assouad dimension is $\ge \alpha$.