How behave the typical $L^q$-dimensions of measures?

TL;DR

This paper investigates the typical behavior of $L^q$-dimensions of probability measures supported on a compact set in $ eal^d$, revealing how these dimensions depend on the set and the parameter $q$.

Contribution

It provides explicit computations of the upper and lower $L^q$-dimensions for typical measures supported on a compact set, linking these to various notions of set dimension across all real $q$.

Findings

Explicit formulas for typical $L^q$-dimensions depending on $q$

Connection between measure dimensions and set dimensions for all $q$

Insights into the behavior of measures supported on fractal-like sets

Abstract

We compute, for a compact set $K\subset\mathbb R^d$, the value of the upper and of the lower $L^q$-dimension of a typical probability measure with support contained in $K$, for any $q\in\mathbb R$. Different definitions of the "dimension" of $K$ are involved to compute these values, following $q\in\mathbb R$.