群作用、Mattila积分及其应用
Group actions, the Mattila integral and applications
摘要：
Mattila积分（$$ {\mathcal M}(μ)=\int {\left( \int_{S^{d-1}} {|\widehatμ(r ω)|}^2 dω\right)}^2 r^{d-1} dr,$$）是研究Falconer距离问题的主要工具。在本文中，我们用一个非常简单的论证，发展了一个广义的版本的马蒂拉积分。我们的第一个应用是考虑距离的乘积$$(Δ(E))^k= \left\{\prod_{j=1}^k |x^j-y^j|: x^j, y^j\in E\right\} $$，并表明，当$d\geq 2$，$(Δ(E))^k$有积极的勒贝格测度，如果$\dim_{\mathcal{H}}(E)>\frac{d}{2}+\frac{1}{4k-1}$。另一个应用是，我们证明了对任何$E,F,H\subset\mathbb{R}^2$，$\dim_{\mathcal{H}}(E)+\dim_{\mathcal{H}}(F)+\dim_{\mathcal{H}}(H)>4$，集合$$E\cdot(F+H)=\{x\cdot(y+z): x\in E, y\in F, z\in H\}$$有正Lebesgue.
英文摘要：
The Mattila integral, $$ {\mathcal M}(μ)=\int {\left( \int_{S^{d-1}} {|\widehatμ(r ω)|}^2 dω\right)}^2 r^{d-1} dr,$$ developed by Mattila, is the main tool in the study of the Falconer distance problem. In this paper, with a very simple argument, we develop a generalized version of the Mattila integral. Our first application is to consider the product of distances $$(Δ(E))^k= \left\{\prod_{j=1}^k |x^j-y^j|: x^j, y^j\in E\right\} $$ and show that when $d\geq 2$, $(Δ(E))^k$ has positive Lebesgue measure if $\dim_{\mathcal{H}}(E)>\frac{d}{2}+\frac{1}{4k-1}$. Another application is, we prove for any $E,F,H\subset\mathbb{R}^2$, $\dim_{\mathcal{H}}(E)+\dim_{\mathcal{H}}(F)+\dim_{\mathcal{H}}(H)>4$, the set $$E\cdot(F+H)=\{x\cdot(y+z): x\in E, y\in F, z\in H\}$$ has positive Lebesgue.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
