平面凸集的整数移位距离集与Fourier基
Distances sets that are a shift of the integers and Fourier basis for planar convex sets
摘要：
本文的目的是证明，如果一个平面集合$A$有一个差集$Δ(A)$满足$Δ(A)\subset \Z^++s$，对于合适的$s$，则$A$最多有3个元素.这个结果是由猜想，磁盘有不超过3个正交指数。进一步，我们证明了：如果$A$是一个关于平面上任意对称凸集$K$相互正交的指数集，且具有光滑边界和处处非零曲率，则有$ # (A \cap {[-q,q]}^2) \leq C(K) q$，其中$C(K)$是只依赖于$K$的常数.这在平面上扩展和澄清了Iosevich和Rudnev的结果。作为推论，我们从\cite{IKP01}和\cite{IKT01}得到了如下结果：如果$K$是具有光滑边界和非零曲率的中心对称凸体，则$L^2(K)$不具有指数正交基.
英文摘要：
The aim of this paper is to prove that if a planar set $A$ has a difference set $Δ(A)$ satisfying $Δ(A)\subset \Z^++s$ 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.
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