Maximality and completeness of orthogonal exponentials on the cube

IF 0.8 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae Pub Date : 2026-02-01 Epub Date: 2025-04-02 DOI:10.1016/j.exmath.2025.125682

Mihail N. Kolountzakis , Nir Lev , Máté Matolcsi

引用次数: 0

Abstract

It is possible to have a packing by translates of a cube that is maximal (i.e. no other cube can be added without overlapping) but does not form a tiling. In the long running analogy of packing and tiling to orthogonality and completeness of exponentials on a domain, we pursue the question whether one can have maximal orthogonal sets of exponentials for a cube without them being complete. We prove that this is not possible in dimensions 1 and 2, but is possible in dimensions 3 and higher. We provide several examples of such maximal incomplete sets of exponentials, differing in size, and we raise relevant questions. We also show that even in dimension 1 there are sets which are spectral (i.e. have a complete set of orthogonal exponentials) and yet they also possess maximal incomplete sets of orthogonal exponentials.

查看原文本刊更多论文

正交指数在立方体上的极大性和完备性

可以通过翻译一个最大的立方体来进行填充（即没有其他立方体可以不重叠地添加），但不形成平铺。在对指数在一个域上的正交性和完备性的长期类比中，我们探讨了一个问题：对于一个立方体，是否可以有指数的极大正交集而不使其完备。我们证明这在1维和2维中是不可能的，但在3维和更高的维中是可能的。我们提供了几个这样的最大不完全指数集的例子，大小不同，并提出了相关的问题。我们还证明了即使在维1中也存在谱集（即具有正交指数的完备集），但它们也具有正交指数的极大不完备集。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Expositiones Mathematicae 数学-数学

CiteScore

1.30

自引率

0.00%

发文量

审稿时长

40 days

期刊介绍： Our aim is to publish papers of interest to a wide mathematical audience. Our main interest is in expository articles that make high-level research results more widely accessible. In general, material submitted should be at least at the graduate level.Main articles must be written in such a way that a graduate-level research student interested in the topic of the paper can read them profitably. When the topic is quite specialized, or the main focus is a narrow research result, the paper is probably not appropriate for this journal. Most original research articles are not suitable for this journal, unless they have particularly broad appeal.Mathematical notes can be more focused than main articles. These should not simply be short research articles, but should address a mathematical question with reasonably broad appeal. Elementary solutions of elementary problems are typically not appropriate. Neither are overly technical papers, which should best be submitted to a specialized research journal.Clarity of exposition, accuracy of details and the relevance and interest of the subject matter will be the decisive factors in our acceptance of an article for publication. Submitted papers are subject to a quick overview before entering into a more detailed review process. All published papers have been refereed.