Variants of the Mattila integral, measures with nonnegative Fourier transforms, and the distance set problem

Abstract:

Suppose $\mu \in M(\mathbb R^n)$ is a measure with $\| \mu \| > 0$, $\sigma$ is surface measure on the unit sphere $\mathbb S^{n-1} \subset \mathbb R^n$, and $\phi \in L^2(\mathbb S^{n-1})$ is a function with $\| \phi \|_{L^2(\mathbb S^{n-1})} > 0$. If $\| \widehat {\mu } \|_{L^2(\mathbb {R}^n)} < \infty$, then $\operatorname {supp} \mu$ has positive Lebesgue measure. We ask the question, what can we say about $\operatorname {supp} \mu$ under the weaker assumption \[ \int _0^\infty \Big | \int _{\mathbb {S}^{n-1}} \widehat {\mu }(r \theta ) \phi (\theta ) d\,\sigma (\theta ) \Big |^2 r^{n-1} \,dr < \infty ? \] We give an answer in the case $\phi \in C^\infty (\mathbb S^{n-1})$ and relate our result to Falconer’s distance set problem. Our line of investigation naturally leads us to the study of measures with nonnegative Fourier transforms, which we also relate to Falconer’s distance set problem in both the Euclidean setting and in vector spaces over finite fields. As an application of our results, we give a new proof of the Erdös-Volkmann ring conjecture.