Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Iterated Spectra of Numbers---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
MathSciNet review: 1333387
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 05D10, 22A15, 54H20, 05B10

Retrieve articles in all journals with MSC (1991): 05D10, 22A15, 54H20, 05B10


Additional Information

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

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

Bryna Kra
Affiliation: Department of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
Email: bryna@math.nuy.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01533-4
PII: S 0002-9947(96)01533-4
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