Dimensions, Embeddings, and Attractors

There is an error in the proof of Corollary 7.8; there is no reason why the functionals $\psi_j$ should exist as claimed.

However, the result $\tau^*(X)\le 2d_{\rm B}(X)$ does hold.

To see this, if $d>d_{\rm B}(X)$ then we can cover $Z:=X-X$ with $\epsilon$-balls centred at $z_j$; there are $\le C\epsilon^{-2d}$ of these.

Now use the Hahn-Banach Theorem to construct linear functions $\psi_j$ with $\|\psi_j\|=1$ and $\psi_j(z_j)=\|z-j\|$; let $V$ be the subsapce of $B^*$ spanned by the $\{\psi_j\}$.

Take $z\in Z$ with $\|z\|\ge 3\epsilon$. Then there exists a $z_k$ such that $\|z-z_k\|<\epsilon$ and so, since $\|z_k\|>2\epsilon$, we have

$|\psi_k(z)|=|\psi_k(z-z_k)+\psi_k(z_k)|\ge\|z_k\|-\|z-z_k\|>\epsilon$.

This shows that $\tau^*(X)\le\sigma_{1/3}(X)\le 2d_{\rm B}(X)$.

In the proof of Theorem 8.1, an attempt to simplify the notation has led to some inaccuracies. $Y$ should be the intersection of $Z_j$ [not $Q_j$] with one of the balls of radius $2^{-j}$, and we should then define

$Q_j(Y)=\{L\in E:\ |(f+L)(z)|\le 2^{-j}\ \mbox{for some }z\in Y\},$

noting that if $L\in Q_j$ then $L\in Q_j(Y)$ for one possible choice of $Y$.

The argument as written bounds $\mu(Q_j(Y))$ [which is what is needed] rather than $\mu(Y)$ [which would make no sense, since $\mu$ is a measure on $E$ and $Y$ is a subset of $\mathscr B$].

Lemma 9.2, it should be $(2^NN^{N/2},N)$ homogeneous. In the second line the cubes should have side $\rho$ and the balls radius $\sqrt{N}\,\rho/2$; the bound on the number of balls is for a cover by balls of radius $\sqrt{N}\,\rho/2$, which leads to the correct result.