Glasner sets and polynomials in primes
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- by R. Nair and S. L. Velani PDF
- Proc. Amer. Math. Soc. 126 (1998), 2835-2840 Request permission
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).References
- N. Alon and Y. Peres, Uniform dilations, Geom. Funct. Anal. 2 (1992), no. 1, 1–28. MR 1143662, DOI 10.1007/BF01895704
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- Shmuel Glasner, Almost periodic sets and measures on the torus, Israel J. Math. 32 (1979), no. 2-3, 161–172. MR 531259, DOI 10.1007/BF02764912
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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
- MR Author ID: 331622
- Email: s.velani@ic.ac.uk
- Received by editor(s): August 19, 1996
- Received by editor(s) in revised form: March 3, 1997
- Communicated by: William W. Adams
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 2835-2840
- MSC (1991): Primary 11K38; Secondary 11K06, 11J13
- DOI: https://doi.org/10.1090/S0002-9939-98-04396-2
- MathSciNet review: 1452815