复Hadamard矩阵与谱集猜想
Complex Hadamard matrices and the Spectral Set Conjecture
摘要：
通过分析复Hadamard矩阵与谱集之间的关系，证明了对任意有限阿贝尔群中大小不超过5的集合A，谱集猜想的方向是“谱->瓦”.然后将这个结果推广到任意维d的无限网格$\Z^d$，最后推广到欧几里得空间。陶最近指出，对于群$\Z_3^5$中的|A| =6，相应的陈述失败了，这个观察很快导致了$\R^5$（陶）和随后的$\R^4$（马托尔奇）中谱集猜想的失败。在本说明的第二部分中，我们进一步降低了这个维度，表明谱集猜想的方向"谱->瓦“在3维中已经是假的。在计算搜索低维（1和2）的反例时，人们至少需要能够有效地判定一个集合是否是一个瓦片（比如说，在一个循环群中）以及它是否是谱的。然而，这种有效的程序是缺乏的，我们提出了一些意见的计算复杂性的一些相关问题。
英文摘要：
By analyzing the connection between complex Hadamard matrices and spectral sets we prove the direction ``spectral -> tile'' of the Sectral Set Conjecture for all sets A of size at most 5 in any finite Abelian group. This result is then extended to the infinite grid $\Z^d$ for any dimension d, and finally to Euclidean space. It was pointed out recently by Tao that the corresponding statement fails for |A|=6 in the group $\Z_3^5$, and this observation quickly led to the failure of the Spectral Set Conjecture in $\R^5$ (Tao), and subsequently in $\R^4$ (Matolcsi). In the second part of this note we reduce this dimension further, showing that the direction ``spectral -> tile'' of the Spectral Set Conjecture is false already in dimension 3. In a computational search for counterexamples in lower dimension (one and two) one needs, at the very least, to be able to decide efficiently if a set is a tile (in, say, a cyclic group) and if it is spectral. Such efficient procedures are lacking however and we make a few comments for the computational complexity of some related problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
