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International Mathematics Research Notices (2003) 2003:2671-2685 , doi:10.1155/S1073792803208126
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Copyright © 2003 Hindawi Publishing Corporation. All rights reserved.

A combinatorial approach to orthogonal exponentials

Alex Iosevich and Mischa Rudnev

We prove that a symmetric strictly convex set with a smooth boundary in Rd can possess no more than finitely many orthogonal exponentials, unless d = 1 mod(4). In such case, the nonexistence theorem is true for a large class of bodies, including the d-dimensional ball. Otherwise, any infinite set of the corresponding exponents necessarily turns out to be a subset of some one-dimensional lattice. We provide examples of convex bodies of revolution in the above dimensions, for which infinite sets of orthogonal exponentials exist. The analysis is reduced to one dimension by studying the distance set of the putative set of exponents with respect to an appropriate metric. A combinatorial principle due to Erdös lies at the heart of the investigation. According to this principle, if the distance set of an infinite set in Rd is a subset of the integers, then the set itself is a subset of some one-dimensional lattice.


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