Abstract:We introduce a threshold parameter $D(\mu)$ for a Borel probability measure $\mu$ with compact support $E\subset\mathbb{R}^n$ such that, for every integer $1\leq m\leq n$, the orthogonal projection of $\mu$ onto a typical $m$ dimensional subspace attains full packing dimension if and only if $m\leq D(\mu)$. In the complementary regime we show that the Assouad dimension of the support controls the possible drop of the packing dimension under projections:$$\dim_P^{m}\mu\geq\dim_P\mu-\max\{0,\ \dim_A E-m\}.$$ 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.
From: Nicolas Angelini [view email]
[v1]
Mon, 20 Apr 2026 13:09:10 UTC (11 KB)
[v2]
Tue, 23 Jun 2026 08:01:06 UTC (11 KB)
[v3]
Fri, 17 Jul 2026 21:08:09 UTC (16 KB)
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