dc.contributor.author | Iosevich, Alex, 1967- | eng |
dc.contributor.author | Jaming, Philippe | eng |
dc.date.issued | 2007 | eng |
dc.description | http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.4133v1.pdf | eng |
dc.description.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. | eng |
dc.identifier.citation | arXiv:0709.4133v1 | eng |
dc.identifier.uri | http://hdl.handle.net/10355/5223 | eng |
dc.language | English | eng |
dc.relation.ispartof | Mathematics publications (MU) | eng |
dc.relation.ispartofcommunity | University of Missouri-Columbia. College of Arts and Sciences. Department of Mathematics | eng |
dc.rights | OpenAccess. | eng |
dc.rights.license | This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. | |
dc.subject | planar sets | eng |
dc.subject.lcsh | Set theory | eng |
dc.title | Distances sets that are a shift of the integers and Fourier basis for planar convex sets | eng |
dc.type | Article | eng |