When the cone in Bochner's theorem has an opening less than π

We prove that if the Fourier transform of a measure μ∈ M (ℝ n ) vanishes in the interior of a cone of opening less than π, then the distance set {| x −τ|: x ∈ G ∩supp μ} has a positive one‐dimensional Lebesgue measure whenever τ∈ℝ n , G ⊂ℝ n is open, and μ( G )≠0. When the cone has an opening greater than π, it is well known that μ is absolutely continuous with respect to Lebesgue measure on ℝ n . For measures on the n ‐torus, this dates back to a 1944 paper of Bochner. For measures on ℝ n , this follows from a Euclidean space version of Forelli's theorem about analytic and quasi‐invariant measures. Forelli's theorem was proved in a 1967 paper in the abstract setting of a locally compact Hausdorff space on which ℝ acts as a topological transformation group. We give a Euclidean space proof of this Euclidean space version of Forelli's theorem.