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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Iterated Spectra of Numbers—Elementary, Dynamical, and Algebraic Approaches
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by Vitaly Bergelson, Neil Hindman and Bryna Kra PDF
Trans. Amer. Math. Soc. 348 (1996), 893-912 Request permission


$IP^*$ 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:
  • Neil Hindman
  • Affiliation: Department of Mathematics, Howard University, Washington, D.C. 20059-0001
  • MR Author ID: 86085
  • Email:
  • Bryna Kra
  • Affiliation: Department of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
  • MR Author ID: 363208
  • ORCID: 0000-0002-5301-3839
  • Email:
  • 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.
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 893-912
  • MSC (1991): Primary 05D10; Secondary 22A15, 54H20, 05B10
  • DOI:
  • MathSciNet review: 1333387