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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Glasner sets and polynomials in primes


Authors: R. Nair and S. L. Velani
Journal: Proc. Amer. Math. Soc. 126 (1998), 2835-2840
MSC (1991): Primary 11K38; Secondary 11K06, 11J13
MathSciNet review: 1452815
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Abstract: A set of integers $S$ is said to be Glasner if for every infinite subset $A$ of the torus $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and $\varepsilon>0$ there exists some $n\in S$ such that the dilation $nA=\{nx\colon\ x\in A\}$ intersects every integral of length $\varepsilon$ in $\mathbb{T}$. In this paper we show that if $p_n$ denotes the $n$th prime integer and $f$ is any non-constant polynomial mapping the natural numbers to themselves, then $(f(p_n))_{n\geq 1}$ is Glasner. The theorem is proved in a quantitative form and generalizes a result of Alon and Peres (1992).


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Additional Information

R. Nair
Affiliation: Department of Pure Mathematics, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, United Kingdom
Email: nair@liv.ac.uk

S. L. Velani
Affiliation: Department of Mathematics, Imperial College, University of London, Huxley Building, 180 Queen’s Gate, London SW7 2BZ, United Kingdom
Email: s.velani@ic.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04396-2
PII: S 0002-9939(98)04396-2
Received by editor(s): August 19, 1996
Received by editor(s) in revised form: March 3, 1997
Communicated by: William W. Adams
Article copyright: © Copyright 1998 American Mathematical Society