Distance sets that are a shift of the integers and Fourier basis for planar convex sets

Abstract

The aim of this paper is to prove that if a planar set A has a difference set Δ(A) satisfying Δ(A) ⊂ ℤ+ + s for suitable s then A has at most 3 elements. This result is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials.Further, we prove that if A is a set of exponentials mutually orthogonal with respect to any symmetric convex set K in the plane with a smooth boundary and everywhere non-vanishing curvature, then #(A ∩ [−q, q]2) ≦ C(K) q where C(K) is a constant depending only on K. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from [8] and [9] that if K is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then L 2(K) does not possess an orthogonal basis of exponentials.