Group actions, the Mattila integral and applications

Abstract:

The Mattila integral, \begin{equation*} {\mathcal M}(\mu )=\int {\left ( \int _{S^{d-1}} {|\widehat {\mu }(r \omega )|}^2 d\omega \right )}^2 r^{d-1} dr, \end{equation*} developed by Mattila, is the main tool in the study of the Falconer distance conjecture. In this paper we develop a generalized version of the Mattila integral that works on more general Falconer-type problems. As applications, we consider when the product of distance set \begin{equation*} (\Delta (E))^k= \left \{\prod _{j=1}^k |x^j-y^j|: x^j, y^j\in E\subset \mathbb {R}^d\right \} \end{equation*} has positive Lebesgue measure and when the sum-product set \begin{equation*} E\cdot (F+H)=\{x\cdot (y+z): x\in E\subset \mathbb {R}^2, y\in F\subset \mathbb {R}^2, z\in H\subset \mathbb {R}^2\}, \end{equation*} has positive Lebesgue.