Let $A$ be a $k$-dimensional rectangle in $\mathbb{R}^k$.
Then $\displaystyle H_p(A)=\sup_{n\in N}H_{p,1/n}(A) \geq \inf_{n \in N}H_{p,1/n}(A)$
How can I find an example (A) such that $H_p(A) > \inf_{n \in N}H_{p,1/n}(A)$
Let $A$ be a $k$-dimensional rectangle in $\mathbb{R}^k$.
Then $\displaystyle H_p(A)=\sup_{n\in N}H_{p,1/n}(A) \geq \inf_{n \in N}H_{p,1/n}(A)$
How can I find an example (A) such that $H_p(A) > \inf_{n \in N}H_{p,1/n}(A)$
In $\mathbb R^1$, consider rectangle $A = [0,1]$ and $p = 1/2$. Then $H_p(A) = \infty$ but $H_{(p,1)}(A)=1$.
It is only a few very special cases when $H_{(p,\epsilon)}(A)$ does not vary with $\epsilon$. Case $p=1$ in $\mathbb R^1$, for example.