Distances 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 $\Delta(A)$ satisfying $\Delta(A)\subset \Z^++s$ 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 \cap {[-q,q]}^2) \leq 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 \cite{IKP01} and \cite{IKT01} 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.