An upper bound for the intermediate dimensions of Bedford–McMullen carpets

Abstract

The intermediate dimensions of a set $\Lambda$, elsewhere denoted by $\dim\_{\theta}\Lambda$, interpolate between its Hausdorff and box dimensions using the parameter $\theta\in\[0,1]$. For a Bedford–McMullen carpet $\Lambda$ with distinct Hausdorff and box dimensions, we show that $\dim\_{\theta}\Lambda$ is strictly less than the box dimension of $\Lambda$ for every $\theta<1$. Moreover, the derivative of the upper bound is strictly positive at $\theta=1$. This answers a question of Fraser; however, determining a precise formula for $\dim\_{\theta}\Lambda$ still remains a challenging problem.