Generalised intermediate dimensions

TL;DR

This paper introduces ta-intermediate dimensions, a new family of dimensions between Hausdorff and box dimensions, extending the theory to broader metric spaces and providing refined geometric insights.

Contribution

It generalizes intermediate dimensions with greater refinement, extends the theory beyond Euclidean spaces, and establishes new bounds and principles for analyzing fractal dimensions.

Findings

ta-intermediate dimensions interpolate between Hausdorff and box dimensions.

The dimensions can recover the interpolation for sets with discontinuous intermediate dimensions.

Provides bounds and principles for dimensions of product sets.

Abstract

We introduce a family of dimensions, which we call the $\Phi$-intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by Falconer, Fraser and Kempton. This is done by restricting the relative sizes of the covering sets in a way that allows for greater refinement than in the definition of the intermediate dimensions. We also extend the theory from Euclidean space to a wider class of metric spaces. We prove that these dimensions can be used to 'recover the interpolation' between the Hausdorff and box dimensions of compact subsets for which the intermediate dimensions are discontinuous at $\theta=0$, thus providing finer geometric information about such sets. We prove continuity-like results involving the Assouad and lower dimensions, which give a sharp general lower bound for the intermediate dimensions that is positive for all $\theta \in (0,1]$ for sets with positive box dimension. We also prove H\"older distortion estimates, a mass distribution principle, and a Frostman type lemma, which we use to study dimensions of product sets.