Fourier dimension of Mandelbrot Cascades on planar curves

TL;DR

This paper proves that Mandelbrot cascades supported on smooth planar curves with curvature have maximal Fourier dimension, matching the infimum of their lower pointwise dimension, revealing their strong Fourier decay properties.

Contribution

It establishes the maximal Fourier dimension for Mandelbrot cascades on curved planar curves, extending understanding of their harmonic analysis properties.

Findings

Fourier dimension equals the infimum of the lower pointwise dimension.

Supports are on $C^2$ curves with nonvanishing curvature.

Maximal Fourier dimension indicates strong decay of Fourier transforms.

Abstract

We consider multifractal Mandelbrot cascades supported on planar $C^2$ curves with nonvanishing curvature and show that their Fourier dimension is as large as possible, i.e., equal to the infimum of the lower pointwise dimension of the measure.