Distances Sets That Are a Shift of the Integers and Fourier Basis for Planar Convex Sets

The aim of this paper is to prove that if a planar set A has a difference set ∆(A) satisfying ∆(A) ⊂ Z + s for suitable s than A has at most 3 elements. This result is motivated by the conjecture that the disk has not 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]) ≤ 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 [IKP01] and [IKT01] that if K is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then L(K) does not possess an orthogonal basis of exponentials.