Fractional Brownian motion and packing dimension (Q1923929)

Let \(X\) be a fractional Brownian motion of index \(\alpha\in]0,1[\), from \(\mathbb{R}\) into \(\mathbb{R}^d\), with \(1>\alpha d\), and let \(\text{Dim }F\) denote the packing dimension of any set \(F\) in \(\mathbb{R}^N\). A compact set \(E\) of \([0,1]\) is constructed, such that \(\text{Dim }X(E)<{1\over\alpha}\text{ Dim }E\); this invalidates a general conjecture on \(\text{Dim }X(E)\). On the other hand, the following general lower bound is given: for any compact set \(E\) in \(\mathbb{R}\), \[ \text{Dim }X(E)\geq {\text{Dim }E\times d\over \alpha d+\text{Dim }E\times(1-\alpha d)}, \] almost surely. And this bound is optimal, being attained for the peculiar \(E\) constructed above.