By Daniel J. Velleman

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Vol. 78, SIAM, Philadelphia, 2008. 6. J. H. Hannay and M. V. Berry, Quantization of linear maps on a torus—Fresnel diffraction by a periodic grating, Physica 1D (1980) 267–290. 7. E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, Dover, New York, 1957. 8. L. Kapitanski and I. Rodnianski, Does a quantum particle know the time? in Emerging Applications of Number Theory, IMA Volumes in Mathematics and Its Applications, vol. 109, Springer Verlag, New York, 1999, 355–371.

I had never seen anything like this before, but the fact that the problem was so elementary convinced me that it must be well known. Nevertheless, all of the leading experts in dispersive waves to whom I showed these computations were similarly surprised, convincing me that I might be onto something new and of potential significance. Some further digging revealed that such rational/irrational behavior had already been noted in the context of the linear Schr¨odinger equation, and leads rapidly into the deep waters of advanced Fourier analysis and exponential sums in number theory.

Moreover, a routine calculation confirms that (6) is true for 0 ≤ k ≤ li + 1. Finally, we have to treat the case i = 5, which is less involved than the cases i ∈ I . Here we directly show that v2k = ti 2k−2 + 2k−2 and v2k+1 = ti 2k−1 + 2k for k ≥ 1, so that we do not have to bother about initial conditions. ) Summing up, we have that the intervals [ξi,1 , ξi,2 ) are disjoint for i = 1, 2, . . , 8 √ √ 2 2 and completely cover [1 − 2 , 2 ). 2. ACKNOWLEDGMENTS. The author is a recipient of an APART-fellowship of the Austrian Academy of Sciences at the University of Waterloo, Canada.

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