关于循环群分解的泛谱和Tijdeman猜想
Universal Spectra and Tijdeman's Conjecture on Factorization of Cyclic Groups
摘要：
R^n中的谱集是有限勒贝格测度的集合X，使得L^2（X）具有指数的正交基。证明了每个谱集R^n都是平移的。一个平移集合T有一个泛谱，如果每个平移T的集合都有这个谱。最近的一个结果证明，许多周期平铺集具有普遍的频谱，使用结果从分解的阿贝尔群，为集团的一个强形式的猜想Tijdeman是有效的。本文证明了Tijdeman猜想对900阶循环群不成立。给出了周期镶嵌集具有泛谱的一个新的充分猜想，并利用它证明了上述反例的镶嵌集具有泛谱。
英文摘要：
A spectral set in R^n is a set X of finite Lebesgue measure such that L^2(X) has an orthogonal basis of exponentials. It is conjectured that every spectral set tiles R^n by translations. A set of translations T has a universal spectrum if every set that that tiles by translations by T has this spectrum. A recent result proved that many periodic tiling sets have universal spectra, using results from factorizations of abelian groups, for groups for which a strong form of a conjecture of Tijdeman is valid. This paper shows Tijdeman's conjecture does not hold for the cyclic group of order 900. It formulates a new sufficient conjecture for a periodic tiling set to have a universal spectrum, and uses it to show that the tiling sets for the counterexample above do have universal spectra.
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