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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Iterated Spectra of Numbersβ€”Elementary, Dynamical, and Algebraic Approaches


Authors: Vitaly Bergelson, Neil Hindman and Bryna Kra
Journal: Trans. Amer. Math. Soc. 348 (1996), 893-912
MSC (1991): Primary 05D10; Secondary 22A15, 54H20, 05B10
DOI: https://doi.org/10.1090/S0002-9947-96-01533-4
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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Additional Information

Vitaly Bergelson
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210-1328
MR Author ID: 35155
Email: vitaly@math.ohio-state.edu

Neil Hindman
Affiliation: Department of Mathematics, Howard University, Washington, D.C. 20059-0001
MR Author ID: 86085
Email: nhindman@aol.com

Bryna Kra
Affiliation: Department of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
MR Author ID: 363208
ORCID: 0000-0002-5301-3839
Email: bryna@math.nuy.ac.il

Received by editor(s): November 5, 1994
Additional Notes: The first two author gratefully acknowledge support received from the National Science Foundation (USA) via grants DMS-9401093 and DMS-9424421 respectively.
Article copyright: © Copyright 1996 American Mathematical Society