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

Vitaly Bergelson, Neil Hindman and Bryna Kra
Transactions of the American Mathematical Society
Vol. 348, No. 3 (Mar., 1996), pp. 893-912
Stable URL: http://www.jstor.org/stable/2155223
Page Count: 20
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## Abstract

IP* sets and central sets are subsets of 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 {[ nα + γ ]: n ∈ N}. Interated 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 {[ nα + γ]: n ∈ 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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