On the Mattila Integral Associated with Sign Indefinite Measures

Abstract

In order to quantitatively illustrate the rôle of positivity in the Falconer distance problem, we construct a family of sign indefinite, compactly supported measures in ${\Bbb R}^d$, such that their Fourier transform and Fourier energy of dimension $s \in (0, d)$ are uniformly bounded. However, the Mattila integral, associated with the Falconer distance problem for these measures is unbounded in the range $0 < s < \frac{d^2}{2d-1}$.