Group actions, the Mattila integral and applications

Abstract

The Mattila integral, $$\mathcal{M}(\mu )=\int {\left({\int }_{S^{d-1}}{|\hat{\mu }(r\omega )|}^{2}d\omega \right)}^2{r}^{d-1}dr,$$ 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 $$(\Delta (E){)}^k=\left\{\prod _{j=1}^k|x^j-y^j|:x^j,y^j\in E\right\}$$ and show that when $d\ge 2$, $(\Delta (E){)}^k$ 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.

manuscript updated

References in corpus (1)

• New results on sum-product type growth over fields