Full packing dimensional projections of measures

Abstract

We introduce a threshold parameter $D(μ)$ for a Borel probability measure $μ$ with compact support $E\subset\mathbb{R}^n$ such that, for every integer $1\leq m\leq n$, the orthogonal projection of $μ$ onto a typical $m$ dimensional subspace attains full packing dimension if and only if $m\leq D(μ)$. In the complementary regime we show that the Assouad dimension of the support controls the possible drop of the packing dimension under projections:$$\dim_P^{m}μ\geq\dim_Pμ-\max\{0,\ \dim_A E-m\}.$$ In particular, whenever $m\geq\dim_A E$, the packing dimension of every measure supported on $E$ is preserved under orthogonal projection onto almost every $m$-dimensional subspace. Taking supremum over the measures supported on a set recovers, in its Assouad dimension form, the corresponding result of Falconer, Fraser and Shmerkin for sets. A key ingredient, of independent interest, is a sharpening of an estimate of Falconer and Mattila for the growth of the measure of balls, in which the ambient dimension is replaced by the Assouad dimension of the support.