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

Abstract

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