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

TL;DR

This paper investigates the structure of planar sets with difference sets shifted by integers and explores Fourier bases for convex sets, establishing bounds on orthogonal exponentials and extending previous results in harmonic analysis.

Contribution

It proves that sets with difference sets contained in shifted positive integers have at most three elements and extends results on orthogonal exponentials for convex sets with smooth boundaries.

Findings

Sets with difference sets in shifted integers have at most 3 elements.

Orthogonal exponentials in certain convex sets are bounded linearly.

L^2 spaces of such convex sets lack orthogonal exponential bases.

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.