# Distance sets that are a shift of the integers and Fourier basis for planar convex sets

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## Abstract

The aim of this paper is to prove that if a planar set *A* has a difference set Δ(*A*) satisfying Δ(*A*) ⊂ ℤ^{+} + *s* for suitable *s* then *A* has at most 3 elements. This result is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials.

Further, we prove that if *A* is a set of exponentials mutually orthogonal with respect to any symmetric convex set *K* in the plane with a smooth boundary and everywhere non-vanishing curvature, then #(*A* ∩ [−*q, q*]^{2}) ≦ *C*(*K*)_{ q } where *C*(*K*) is a constant depending only on *K*. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from [8] and [9] that if *K* is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then *L* ^{2}(*K*) does not possess an orthogonal basis of exponentials.

## Key words and phrases

distance sets orthogonal exponentials convex sets## 2000 Mathematics Subject Classification

42B05 42C30 05B10## Preview

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