Generalised intermediate dimensions

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 is an element of (0, 1] for sets with positive box dimension. We also prove Holder distortion estimates, a mass distribution principle, and a Frostman type lemma, which we use to study dimensions of product sets.

Automated summary

We introduce a family of dimensions called the Phi-intermediate dimensions, which generalize the intermediate dimensions introduced by Falconer, Fraser and Kempton and lie between the Hausdorff and box dimensions. This is achieved by restricting the relative sizes of the covering sets to allow for greater refinement. We extend the theory from Euclidean space to a wider class of metric spaces and prove that these dimensions can be used to recover the interpolation between the Hausdorff and box dimensions of compact subsets.