{"title":"凸域的Fuglede猜想在所有维度上都是正确的","authors":"Nir Lev, M. Matolcsi","doi":"10.4310/ACTA.2022.v228.n2.a3","DOIUrl":null,"url":null,"abstract":"A set $\\Omega \\subset \\mathbb{R}^d$ is said to be spectral if the space $L^2(\\Omega)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $\\Omega$ is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body $\\Omega \\subset \\mathbb{R}^d$ the \"tiling implies spectral\" part of the conjecture is in fact true. \nTo the contrary, the \"spectral implies tiling\" direction of the conjecture for convex bodies was proved only in $\\mathbb{R}^2$, and also in $\\mathbb{R}^3$ under the a priori assumption that $\\Omega$ is a convex polytope. In higher dimensions, this direction of the conjecture remained completely open (even in the case when $\\Omega$ is a polytope) and could not be treated using the previously developed techniques. \nIn this paper we fully settle Fuglede's conjecture for convex bodies affirmatively in all dimensions, i.e. we prove that if a convex body $\\Omega \\subset \\mathbb{R}^d$ is a spectral set then it can tile the space by translations. To prove this we introduce a new technique, involving a construction from crystallographic diffraction theory, which allows us to establish a geometric \"weak tiling\" condition necessary for a set $\\Omega \\subset \\mathbb{R}^d$ to be spectral.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":4.9000,"publicationDate":"2019-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"44","resultStr":"{\"title\":\"The Fuglede conjecture for convex domains is true in all dimensions\",\"authors\":\"Nir Lev, M. Matolcsi\",\"doi\":\"10.4310/ACTA.2022.v228.n2.a3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A set $\\\\Omega \\\\subset \\\\mathbb{R}^d$ is said to be spectral if the space $L^2(\\\\Omega)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $\\\\Omega$ is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body $\\\\Omega \\\\subset \\\\mathbb{R}^d$ the \\\"tiling implies spectral\\\" part of the conjecture is in fact true. \\nTo the contrary, the \\\"spectral implies tiling\\\" direction of the conjecture for convex bodies was proved only in $\\\\mathbb{R}^2$, and also in $\\\\mathbb{R}^3$ under the a priori assumption that $\\\\Omega$ is a convex polytope. In higher dimensions, this direction of the conjecture remained completely open (even in the case when $\\\\Omega$ is a polytope) and could not be treated using the previously developed techniques. \\nIn this paper we fully settle Fuglede's conjecture for convex bodies affirmatively in all dimensions, i.e. we prove that if a convex body $\\\\Omega \\\\subset \\\\mathbb{R}^d$ is a spectral set then it can tile the space by translations. To prove this we introduce a new technique, involving a construction from crystallographic diffraction theory, which allows us to establish a geometric \\\"weak tiling\\\" condition necessary for a set $\\\\Omega \\\\subset \\\\mathbb{R}^d$ to be spectral.\",\"PeriodicalId\":50895,\"journal\":{\"name\":\"Acta Mathematica\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":4.9000,\"publicationDate\":\"2019-04-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"44\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/ACTA.2022.v228.n2.a3\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/ACTA.2022.v228.n2.a3","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
The Fuglede conjecture for convex domains is true in all dimensions
A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $\Omega$ is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body $\Omega \subset \mathbb{R}^d$ the "tiling implies spectral" part of the conjecture is in fact true.
To the contrary, the "spectral implies tiling" direction of the conjecture for convex bodies was proved only in $\mathbb{R}^2$, and also in $\mathbb{R}^3$ under the a priori assumption that $\Omega$ is a convex polytope. In higher dimensions, this direction of the conjecture remained completely open (even in the case when $\Omega$ is a polytope) and could not be treated using the previously developed techniques.
In this paper we fully settle Fuglede's conjecture for convex bodies affirmatively in all dimensions, i.e. we prove that if a convex body $\Omega \subset \mathbb{R}^d$ is a spectral set then it can tile the space by translations. To prove this we introduce a new technique, involving a construction from crystallographic diffraction theory, which allows us to establish a geometric "weak tiling" condition necessary for a set $\Omega \subset \mathbb{R}^d$ to be spectral.