Additive properties of fractal sets on the parabola

Abstract

Let $0\le s\le 1$, 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 ${\dim }_{\mathrm{H}}(K+K)\ge 2s$. The main corollary of the paper states that if $0<s<1$, then adding $K$ once more makes the sum slightly larger: $${\dim }_{\mathrm{H}}(K+K+K)\ge 2s+\epsilon ,$$ 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 $\mu (B(x,r))\le r^s$ for all $x\in \mathbb{P}$ and $r>0$, then there exists $\epsilon =\epsilon (s)>0$ such that $$\Vert \hat{\mu }{\Vert }_{L^6(B(R))}^6\le R^{2-(2s+\epsilon )}$$ for all sufficiently large $R\ge 1$. The proof is based on a reduction to a $\delta$-discretised point-circle incidence problem, and eventually to the $(s,2s)$-Furstenberg set problem.

arXiv comment26 pages, 2 figures