$L^2$-Flattening of Self-similar Measures on Non-degenerate Curves

TL;DR

This paper proves that self-similar measures on curves in Euclidean space have Fourier transforms with controlled $L^p$ norms, leading to improved $L^2$-dimension through convolution, advancing understanding of fractal measures.

Contribution

It establishes $L^2$-flattening results for self-similar measures on non-degenerate curves, connecting Fourier decay with dimension improvement.

Findings

Fourier transform of the measure has $L^p$ bounds for all $p>1$

Convolution with the measure enhances $L^2$-dimension

Provides quantitative Fourier decay estimates for self-similar measures

Abstract

Let $\mu$ be a non-atomic self-similar measure on $\mathbb{R}$, and let $\nu$ be its pushforward to a non-degenerate curve in $\mathbb{R}^d, d\geq 1$. We show that for every $\epsilon>0$, there is $p>1$, so that $\left \lVert \hat{\nu} \right \rVert_{L^p(B(R))}^p = O_\epsilon(R^\epsilon)$ for all $R>1$, where $B(R)$ is the $R$-ball about the origin. As a corollary, we show that convolution with $\nu$ quantitatively improves $L^2$-dimension.