Attainable forms of intermediate dimensions

Abstract

The intermediate dimensions are a family of dimensions which interpolate between the Hausdorff and box dimensions of sets. We prove a necessary and sufficient condition for a given function h(θ) to be realized as the intermediate dimensions of a bounded subset of R d . This condition is a straightforward constraint on the Dini derivatives of h(θ), which we prove is sharp using a homogeneous Moran set construction.

Intermediate dimensions of infinitely generated attractors

“…In this paper we study the intermediate dimensions of limit sets of infinite iterated function systems. The intermediate dimensions have been studied further in [2,3,4,5,6,10,13,14,21,41] and have been generalised to the Φ-intermediate dimensions by Banaji [1] to give more refined geometric information about sets for which the intermediate dimensions are discontinuous at θ = 0, which by Theorem 3.5 can happen for the limit sets studied in this paper (see the discussion after Theorem 4.3). The intermediate dimensions are an example of a broader notion of 'dimension interpolation' (see the survey [21]), which seeks to find a geometrically natural family of dimensions which lie between two familiar notions of dimension.…”

“…This is in contrast to sets whose intermediate dimensions have previously been calculated such as spirals [6], sequences [17,Section 3.1], and concentric spheres and topologist's sine curves [41], where only the two extreme scales were used in the cover. Recently, it has been shown that more than two scales are needed to attain the intermediate dimensions certain inhomogeneous Moran sets (see [3,Remark 3.11]) and Bedford-McMullen carpets for small values of θ (see [2,Corollary 2.2]). In the conformal setting, Mauldin and Urbański [35,36] proved results for the Hausdorff and upper box dimensions.…”