Let \(E\subset\mathbb{R}^n\) and \(F\subset\mathbb{R}^m\) be an \(s\)-set and a \(t\)-set, respectively. The authors prove, under some conditions on the lower densities of \(E\) and \(F\), that \(\dim_{\text{H}}(E\times F)= \dim_{\text{H}} E+\dim_{\text{H}} F\), where \(\dim_{\text{H}}\) denotes the Hausdorff dimension.
Since the conditions of this paper imply \(\dim_{\text{H}} E= \dim_{\text{P}}E\), where \(\dim_{\text{P}}\) denotes the packing dimension, the authors’ results also follow from the following inequality of C. Tricot jun. [Math. Proc. Camb. Philos. Soc. 91, 57-74 (1982; Zbl 0483.28010)]: \(\dim_{\text{H}} E+ \dim_{\text{H}} F\leq\dim_{\text{H}} (E\times F)\leq \dim_{\text{P}} E+ \dim_{\text{H}} F\).