Generalised intermediate dimensions

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