论文标题
抛物线上的分形集的添加特性
Additive properties of fractal sets on the parabola
论文作者
论文摘要
令$ 0 \ leq s \ leq 1 $，然后让$ \ mathbb {p}：= \ {（t，t，t，t^{2}）\ in \ mathbb {r}^{2}^{2}：t \ in [-1,1] \} $。如果$ k \ subset \ mathbb {p} $是一个封闭的集合，带有$ \ dim _ {\ mathrm {h}} k = s $，则不难看到$ \ dim _ {\ mathrm {h h}}（k + k + k）\ geq 2s $。本文的主要推论指出，如果$ 0 <s <1 $，那么再增加$ k $，使总和稍大：$ \ dim _ {\ mathrm {h}}}（k + k + k + k）\ geq 2s +ε，$ε，$ε=ε（s）> 0 $。此信息是从$ \ Mathbb {p} $上的Frostman测量的傅立叶变换中绑定的$ l^{6} $推导的。如果$ 0 <s <1 $，而$μ$是$ \ mathbb {p} $满足$μ（b（x，r））\ leq r^{s} $的borel量度\ | \hatμ\ | _ {l^{6}（b（r））}^{6} \ leq r^{2 - （2S +ε）} $$对于所有足够大的$ r \ geq 1 $。证明是基于减少$δ$ distcretist-circle circle发病率问题的基础，最终是$（s，2s）$ - furstenberg设定的问题。
Let $0 \leq s \leq 1$, and let $\mathbb{P} := \{(t,t^{2}) \in \mathbb{R}^{2} : t \in [-1,1]\}$. If $K \subset \mathbb{P}$ is a closed set with $\dim_{\mathrm{H}} K = s$, it is not hard to see that $\dim_{\mathrm{H}} (K + K) \geq 2s$. The main corollary of the paper states that if $0 < s < 1$, then adding $K$ once more makes the sum slightly larger: $$\dim_{\mathrm{H}} (K + K + K) \geq 2s + ε, $$ where $ε= ε(s) > 0$. This information is deduced from an $L^{6}$ bound for the Fourier transforms of Frostman measures on $\mathbb{P}$. If $0 < s < 1$, and $μ$ is a Borel measure on $\mathbb{P}$ satisfying $μ(B(x,r)) \leq r^{s}$ for all $x \in \mathbb{P}$ and $r > 0$, then there exists $ε= ε(s) > 0$ such that $$ \|\hatμ\|_{L^{6}(B(R))}^{6} \leq R^{2 - (2s + ε)} $$ for all sufficiently large $R \geq 1$. The proof is based on a reduction to a $δ$-discretised point-circle incidence problem, and eventually to the $(s,2s)$-Furstenberg set problem.
