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.