Strongly central sets and sets of polynomial returns mod 1

Authors:
Vitaly Bergelson, Neil Hindman and Dona Strauss

Journal:
Proc. Amer. Math. Soc. **140** (2012), 2671-2686

MSC (2010):
Primary 05D10

DOI:
https://doi.org/10.1090/S0002-9939-2011-11129-8

Published electronically:
December 22, 2011

MathSciNet review:
2910755

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Abstract | References | Similar Articles | Additional Information

Abstract: *Central* sets in were introduced by Furstenberg and are known to have substantial combinatorial structure. For example, any central set contains arbitrarily long arithmetic progressions, all finite sums of distinct terms of an infinite sequence, and solutions to all partition regular systems of homogeneous linear equations. We introduce here the notions of *strongly central* and *very strongly central*, which as the names suggest are strictly stronger than the notion of central. They are also strictly stronger than *syndetic*, which in the case of means that gaps are bounded.

Given , let . Kronecker's Theorem says that if are linearly independent over and is a nonempty open subset of , then is nonempty, and Weyl showed that this set has positive density. We show here that if is in the closure of , then this set is strongly central. More generally, let be real polynomials with zero constant term. We show that

**1.**J. Auslander,*Minimal flows and their extensions*, North Holland, Amsterdam, 1988. MR**956049 (89m:54050)****2.**V. Bergelson,*Ergodic Ramsey Theory - an update*, London Math. Soc. Lecture Note Series**228**, Cambridge University Press (1996), 1-61. MR**1411215 (98g:28017)****3.**V. Bergelson,*Minimal idempotents and ergodic Ramsey Theory*, London Math. Soc. Lecture Note Series**310**, Cambridge University Press (2003), 8-39. MR**2052273 (2006b:37022)****4.**V. Bergelson,*Ultrafilters, IP sets, dynamics, and combinatorial number theory*, Contemporary Math.**530**, Amer. Math. Soc. (2010), 23-47. MR**2757532****5.**V. Bergelson and N. Hindman,*Nonmetrizable topological dynamics and Ramsey Theory*, Trans. Amer. Math. Soc.**320**(1990), 293-320. MR**982232 (90k:03046)****6.**V. Bergelson, N. Hindman, and R. McCutcheon,*Notions of size and combinatorial properties of quotient sets in semigroups*, Topology Proceedings**23**(1998), 23-60. MR**1743799 (2001a:20114)****7.**V. Bergelson and A. Leibman,*Distribution of values of bounded generalized polynomials*, Acta Mathematica**198**(2007), 155-230. MR**2318563 (2008m:11149)****8.**C. Chou,*On a geometric property of the set of invariant means on a group*, Proc. Amer. Math. Soc.**30**(1971), 296-302. MR**0283584 (44:815)****9.**T. Carlson, N. Hindman, J. McLeod, and D. Strauss,*Almost disjoint large subsets of semigroups*, Topology and its Applications**155**(2008), 433-444. MR**2380928 (2008m:54050)****10.**D. De, N. Hindman, and D. Strauss,*A new and stronger Central Sets Theorem*, Fund. Math.**199**(2008), 155-175. MR**2410923 (2009c:05248)****11.**R. Ellis,*Distal transformation groups*, Pacific J. Math.**8**(1958), 401-405. MR**0101283 (21:96)****12.**H. Furstenberg,*Recurrence in ergodic theory and combinatorical number theory*, Princeton University Press, Princeton, NJ, 1981. MR**603625 (82j:28010)****13.**G. Hardy and J. Littlewood,*Some problems of Diophantine approximation*, Acta. Math.**37**(1914), 155-191. MR**1555098****14.**N. Hindman, I. Leader, and D. Strauss,*Infinite partition regular matrices - solutions in central sets*, Trans. Amer. Math. Soc.**355**(2003), 1213-1235. MR**1938754 (2003h:05187)****15.**N. Hindman and D. Strauss,*Algebra in the Stone-Čech compactification: theory and applications*, de Gruyter, Berlin, 1998. MR**1642231 (99j:54001)****16.**L. Kronecker,*Die Periodensysteme von Funktionen Reeller Variablen*, Berliner Situngsberichte (1884), 1071-1080.**17.**L. Kuipers and H. Niederreiter,*Uniform distribution of sequences*, Wiley, New York, 1974. MR**0419394 (54:7415)****18.**R. McCutcheon,*Elemental methods in ergodic Ramsey theory*, Springer-Verlag, Berlin, 1999. MR**1738544 (2001c:05141)****19.**H. Shi and H. Yang,*Nonmetrizable topological dynamical characterization of central sets*, Fund. Math.**150**(1996), 1-9. MR**1387952 (97j:54045)****20.**H. Weyl,*Über die Gleichverteilung von Zahlen mod eins*, Math. Ann.**77**(1916), 313-352. MR**1511862**

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Additional Information

**Vitaly Bergelson**

Affiliation:
Department of Mathematics, Ohio State University, Columbus, Ohio 43210

Email:
vitaly@math.ohio-state.edu

**Neil Hindman**

Affiliation:
Department of Mathematics, Howard University, 2400 Sixth Street, NW, Washington, DC 20059

Email:
nhindman@aol.com

**Dona Strauss**

Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9J2, United Kingdom

Email:
d.strauss@hull.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-2011-11129-8

Received by editor(s):
June 17, 2010

Received by editor(s) in revised form:
March 16, 2011

Published electronically:
December 22, 2011

Additional Notes:
The first two authors acknowledge support received from the National Science Foundation via Grants DMS-0901106 and DMS-0852512 respectively.

Communicated by:
Jim Haglund

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.