Packing dimension and Cartesian products

Abstract

We show that for any analytic set A A in R d \mathbf {R}^d , its packing dimension dim P ⁡ ( A ) \dim _{\mathrm {P}}(A) can be represented as sup B { dim H ⁡ ( A × B ) − dim H ⁡ ( B ) } , \; \sup _B \{ \dim _{\mathrm {H}} (A \times B) -\dim _{\mathrm {H}}(B) \} \, , \, where the supremum is over all compact sets B B in R d \mathbf {R}^d , and dim H \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 P ⁡ ( A ) > d \dim _{\mathrm {P}} (A) > d . In contrast, we show that the dual quantity inf B { dim P ⁡ ( A × B ) − dim P ⁡ ( B ) } , \; \inf _B \{ \dim _{\mathrm {P}}(A \times B) -\dim _{\mathrm {P}}(B) \} \, , \, is at least the “lower packing dimension” of A A , but can be strictly greater. (The lower packing dimension is greater than or equal to the Hausdorff dimension.)