Packing dimension and Cartesian products

Abstract:

We show that for any analytic set $A$ in $\mathbf {R}^d$, its packing dimension $\dim _{\mathrm {P}}(A)$ can be represented as $\; \sup _B \{ \dim _{\mathrm {H}} (A \times B) -\dim _{\mathrm {H}}(B) \} ,$ where the supremum is over all compact sets $B$ in $\mathbf {R}^d$, and $\dim _{\mathrm {H}}$ denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if $\dim _{\mathrm {P}} (A) < d$. In contrast, we show that the dual quantity $\; \inf _B \{ \dim _{\mathrm {P}}(A \times B) -\dim _{\mathrm {P}}(B) \} ,$ is at least the “lower packing dimension” of $A$, but can be strictly greater. (The lower packing dimension is greater than or equal to the Hausdorff dimension.)