Additive properties of fractal sets on the parabola

TL;DR

This paper investigates the additive properties of fractal sets on the parabola, showing that triple sums increase Hausdorff dimension slightly, using Fourier analysis and incidence geometry techniques.

Contribution

It establishes a new lower bound on the Hausdorff dimension of triple sums of fractal sets on the parabola, improving understanding of their additive structure.

Findings

Triple sums of fractal sets on the parabola have Hausdorff dimension at least 2s + ε.

An L^6 Fourier transform bound for Frostman measures on the parabola is proven.

The proof reduces to a point-circle incidence problem and Furstenberg set analysis.

Abstract

Let $0 \leq s \leq 1$, and let $\mathbb{P} := \{(t,t^{2}) \in \mathbb{R}^{2} : t \in [-1,1]\}$. If $K \subset \mathbb{P}$ is a closed set with $\dim_{\mathrm{H}} K = s$, it is not hard to see that $\dim_{\mathrm{H}} (K + K) \geq 2s$. The main corollary of the paper states that if $0 < s < 1$, then adding $K$ once more makes the sum slightly larger: $$\dim_{\mathrm{H}} (K + K + K) \geq 2s + \epsilon, $$ where $\epsilon = \epsilon(s) > 0$. This information is deduced from an $L^{6}$ bound for the Fourier transforms of Frostman measures on $\mathbb{P}$. If $0 < s < 1$, and $\mu$ is a Borel measure on $\mathbb{P}$ satisfying $\mu(B(x,r)) \leq r^{s}$ for all $x \in \mathbb{P}$ and $r > 0$, then there exists $\epsilon = \epsilon(s) > 0$ such that $$ \|\hat{\mu}\|_{L^{6}(B(R))}^{6} \leq R^{2 - (2s + \epsilon)} $$ for all sufficiently large $R \geq 1$. The proof is based on a reduction to a $\delta$-discretised point-circle incidence problem, and eventually to the $(s,2s)$-Furstenberg set problem.