Hausdorff dimension of multidimensional multiplicative subshifts

Abstract

AbstractThe purpose of this study is two-fold. First, the Hausdorff dimension formula of the multidimensional multiplicative subshift (MMS) in $\mathbb {N}^d$ is presented. This extends the earlier work of Kenyon et al [Hausdorff dimension for fractals invariant under multiplicative integers. Ergod. Th. & Dynam. Sys.32(5) (2012), 1567–1584] from $\mathbb {N}$ to $\mathbb {N}^d$ . In addition, the preceding work of the Minkowski dimension of the MMS in $\mathbb {N}^d$ is applied to show that their Hausdorff dimension is strictly less than the Minkowski dimension. Second, the same technique allows us to investigate the multifractal analysis of multiple ergodic average in $\mathbb {N}^d$ . Precisely, we extend the result of Fan et al, [Multifractal analysis of some multiple ergodic averages. Adv. Math.295 (2016), 271–333] of the multifractal analysis of multiple ergodic average from $\mathbb {N}$ to $\mathbb {N}^d$ .