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The Relationship Between $\epsilon$-Kronecker Sets and Sidon Sets

  Published:2016-01-29
 Printed: Sep 2016
  • Kathryn Hare,
    Dept. of Pure Mathematics, University of Waterloo, Waterloo, Ont.,~Canada, N2L 3G1
  • L. Thomas Ramsey,
    Dept. of Mathematics, University of Hawaii , Honolulu, HI 96822
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Abstract

A subset $E$ of a discrete abelian group is called $\epsilon $-Kronecker if all $E$-functions of modulus one can be approximated to within $\epsilon $ by characters. $E$ is called a Sidon set if all bounded $E$-functions can be interpolated by the Fourier transform of measures on the dual group. As $% \epsilon $-Kronecker sets with $\epsilon \lt 2$ possess the same arithmetic properties as Sidon sets, it is natural to ask if they are Sidon. We use the Pisier net characterization of Sidonicity to prove this is true.
Keywords: Kronecker set, Sidon set Kronecker set, Sidon set
MSC Classifications: 43A46, 42A15, 42A55 show english descriptions Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
Trigonometric interpolation
Lacunary series of trigonometric and other functions; Riesz products
43A46 - Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
42A15 - Trigonometric interpolation
42A55 - Lacunary series of trigonometric and other functions; Riesz products
 

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