Spatially independent martingales, intersections, and applications

TL;DR

This paper introduces spatially independent martingales, a new class of random measures, and demonstrates their geometric properties, including intersection regularity, projections, convolutions, and Fourier decay, unifying and extending existing fractal measure results.

Contribution

It defines spatially independent martingales and proves their intersection measures are Hölder continuous, leading to broad generalizations of fractal geometry theorems.

Findings

Hölder continuity of intersection measures as a function of parameters

Enhanced Marstrand-Mattila projection and slicing results

Smoothness of measure convolutions and sumset interior

Abstract

We define a class of random measures, spatially independent martingales, which we view as a natural generalisation of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrised measures $\{\eta_t\}_t$, and show that under some natural checkable conditions, a.s. the total measure of the intersections is H\"older continuous as a function of $t$. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals we establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, (d) rapid Fourier decay. Among other applications, we obtain an answer to a question of I. {\L}aba in connection to the restriction problem for fractal measures.