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A Discrete Fourier Kernel and Fraenkel's Tiling Conjecture

Ron Graham·Kevin O'Bryant
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Abstract

The set B_{p,r}^q:=\{\floor{nq/p+r} \colon n\in Z \} with integers p, q, r) is a Beatty set with density p/q. We derive a formula for the Fourier transform \hat{B_{p,r}^q}(j):=\sum_{n=1}^p e^{-2 πi j \floor{nq/p+r} / q}. A. S. Fraenkel conjectured that there is essentially one way to partition the integers into m>2 Beatty sets with distinct densities. We conjecture a generalization of this, and use Fourier methods to prove several special cases of our generalized conjecture.

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BibTeXRIS

Ron Graham, Kevin O'Bryant. 2005-02-28. A Discrete Fourier Kernel and Fraenkel's Tiling Conjecture. https://doi.org/10.4064/aa118-3-4

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