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

  • István Kolossváry

    University of St Andrews, UK
An upper bound for the intermediate dimensions of Bedford–McMullen carpets cover
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Abstract

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