2026/04/20 by Nicolas Angelini
Mathematics · #math.CA #math.PR
We introduce a threshold parameter D(μ) for a Borel probability measure μ with compact support E⊂ℝn such that, for every integer 1≤ m≤ n, the orthogonal projection of μ onto a typical m dimensional subspace attains full packing dimension if and only if m≤ 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:dimPmμ≥dimPμ-max\0, dimA E-m\. In particular, whenever m≥dimA 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.