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 \(0 < s < \frac{d^2}{2d-1}\).
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