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

### István Kolossváry

University of St Andrews, UK

## 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.

## Cite this article

István Kolossváry, An upper bound for the intermediate dimensions of Bedford–McMullen carpets. J. Fractal Geom. 9 (2022), no. 1/2, pp. 151–169

DOI 10.4171/JFG/118