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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
 Format: LaTeX MathJax PDF

## 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
 MSC Classifications: 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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