A set omega subset of Double-struck capital R-2 is said to be spectral if the space L-2(omega) has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets "behave like" sets which can tile the space by translations. This suggests a conjecture that a product set omega = A x B is spectral if and only if the factors A and B are both spectral sets. We recently proved this in the case when A is an interval in dimension one. The main result of the present paper is that the conjecture is true also when A is a convex polygon in two dimensions. We discuss this result in connection with the conjecture that a convex polytope omega is spectral if and only if it can tile by translations.
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机译：如果空间L-2（OMEGA）具有指数函数的正交基础，则据说一个设置的双击资资本R-2的omega子集是频谱。 众所周知，在许多方面，频谱集“表现得像”这样的集合，其可以通过翻译铺设空间。 这表明猜测产品集OMEGA = A X B是频谱IF且仅当A和B都是光谱集时。 当A是维度一个的间隔时，我们最近证明了这一点。 本文的主要结果是，当A是两个维度的凸多边形时，猜想也是如此。 我们讨论该结果与猜想鉴于凸起的Polytope Omega是频谱，如果它只能通过翻译铺设。
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