Furstenberg set theorem for transversal families of functions

TL;DR

This paper extends the Furstenberg set theorem to transversal families of functions and applies it to bound Fourier transforms of fractal measures on plane curves.

Contribution

It introduces a new extension of the Furstenberg set theorem for transversal function families, with applications to Fourier analysis of fractal measures.

Findings

Extended Furstenberg set theorem to transversal families

Derived bounds on L^p-norms of Fourier transforms of fractal measures

Applicable to measures supported on plane curves

Abstract

We prove an extension of the Furstenberg set theorem to families of graphs satisfying a transversality condition. We apply the result to derive bounds on $L^{p}$-norms of Fourier transforms of fractal measures supported on plane curves.