AI 中文总结
研究富格莱德猜想在二维空间的情况，通过在二阶有限阿贝尔群中构造特定子集，利用有限到无限转移原理得到\(\R^2\)中有界子集，证明该猜想在二维空间中两个方向都不成立。
AI 中文摘要
富格莱德猜想断言，一个具有正的有限测度的可测集是谱集当且仅当它通过平移平铺欧几里得空间。已知在每个维度\(d\geq3\)中存在反例，而一维和二维情况仍未解决。我们在二阶有限阿贝尔群\(\Z_{60}\times\Z_{12}\)中构造了两个明确的\(60\)点子集：一个是没有谱的平移平铺集，另一个是谱集但不平铺。有限到无限的转移原理将它们提升到\(\R^2\)中的有界子集，这些子集是单位正方形的有限并。因此，富格莱德猜想在二维空间中的两个蕴含关系都不成立。
英文摘要
Fuglede's conjecture asserts that a measurable set of positive and finite measure is spectral if and only if it tiles Euclidean space by translations. Counterexamples are known in every dimension $d\ge3$, whereas the one- and two-dimensional cases have remained unresolved. We construct two explicit $60$-point subsets of the rank-two finite Abelian group $\Z_{60}\times\Z_{12}$: one is a translational tile with no spectrum, and the other is spectral but does not tile. A finite-to-infinite transference principle lifts them to bounded subsets of $\R^2$ that are finite unions of unit squares. Consequently, both implications in Fuglede's conjecture fail in dimension two.
Comments18 pages
