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ISSN 1088-6850(e) ISSN 0002-9947(p)

Iterated Spectra of Numbers---Elementary, Dynamical, and Algebraic Approaches

Author(s): Vitaly Bergelson; Neil Hindman; Bryna Kra
Journal: Trans. Amer. Math. Soc. 348 (1996), 893-912.
MSC (1991): Primary 05D10; Secondary 22A15, 54H20, 05B10
MathSciNet review: 1333387
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Abstract: $IP^*$ sets and central sets are subsets of $\mathbb N$ which arise out of applications of topological dynamics to number theory and are known to have rich combinatorial structure. Spectra of numbers are often studied sets of the form $\{[n\alpha+\gamma]\colon n\in\mathbb N\}$ . Iterated spectra are similarly defined with $n$ coming from another spectrum. Using elementary, dynamical, and algebraic approaches we show that iterated spectra have significantly richer combinatorial structure than was previously known. For example we show that if $\alpha>0$ and $0<\gamma<1$ , then $\{[n\alpha+\gamma]\colon n\in\mathbb N\}$ is an $IP^*$ set and consequently contains an infinite sequence together with all finite sums and products of terms from that sequence without repetition.

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