Abstract

The Fuglede conjecture states that a set is spectral if and only if it tiles by translation. The conjecture was disproved by Tao for dimensions 5 and higher by giving a counterexample in ${\mathbb{Z}}_3^5.$ We present a computer program that determines that the Fuglede conjecture holds in ${\mathbb{Z}}_3^5$ by exhausting the search space. Recently Iosevich, Mayeli and Pakianathan showed that the Fuglede conjecture holds over prime fields when the dimension does not exceed 2. The question for dimension 3 was previously addressed by Aten et al. for p = 3. In this paper we build upon those results by Aten et al. to allow a computer to carry out the lengthy computations.

摘要

Fuglede 猜想指出，一个集合是光谱的当且仅当它通过平移平铺。陶在维度 5 及更高维度上通过给出反例来反驳该猜想${\mathbb{Z}}_3^5.$我们提出了一个计算机程序来确定 Fuglede 猜想在${\mathbb{Z}}_3^5$通过耗尽搜索空间。最近 Iosevich、Mayeli 和 Pakianathan 表明，当维数不超过 2 时，Fuglede 猜想在素数场上成立。维数 3 的问题之前由 Aten 等人解决。对于p  = 3。在本文中，我们以 Aten 等人的这些结果为基础。允许计算机执行冗长的计算。