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Acta Mathematica Hungarica

, Volume 121, Issue 1–2, pp 107–118 | Cite as

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

  • A. IosevichEmail author
  • P. Jaming
Article

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 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Missouri-ColumbiaColumbiaUSA
  2. 2.MAPMO-Fédération Denis PoissonUniversité d’OrléansOrleans Cedex 2France

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