For a positive Radon measure \(\mu\) in \({\mathbb{R}}^ n \)with compact support, let \(\sigma\) (\(\mu)\)(r) be the average of \(| {\hat \mu}|\) 2 over the sphere \(\{x:| x| =r\}\), and let \(I_ s(\mu)\) be the energy-integral \(\iint | x-y|^{-s} d\mu x d\mu y.\) It is shown that for \(0<s\leq (n-1)/2,\sigma (\mu)(r)\leq cI_ s(\mu)r^{-s}\) and that for \((n-1)/2<s<n\) only weaker estimates hold. Such relations between \(\sigma\) (\(\mu)\) and \(I_ s(\mu)\) are used to study the Hausdorff dimension of the distance sets \(\{| x-y|: x\in A,\quad y\in B\}\) and the intersections \(A\cap fB\), where f runs through the isometries of \({\mathbb{R}}^ n,\) and A and B are Borel sets in \({\mathbb{R}}^ n.\) This continues the earlier work of K. J. Falconer [Mathematika 32, 206-212 (1985; Zbl 0605.28005)] and the author [ibid. 213-217 (1985; Zbl 0569.28008)].