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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-96-01533-4
MathSciNet review: 1333387
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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.


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  • 1. J. Auslander On the proximal relation in topological dynamics, Proc. Amer. Math. Soc. 11 (1960), 890--895. MR 29:1632
  • 2. J. Baker and P. Milnes, The ideal structure of the Stone-\v{C}ech compactification of a group, Math. Proc. Cambridge Philos. Soc. 82 (1977), 401--409. MR 57:509
  • 3. T. Bang, On the sequence $[n\alpha]$, $n=1,2,\dots$, Math. Scand. 5 (1957), 69--76. MR 19:1159h
  • 4. V. Bergelson and N. Hindman, A combinatorially large cell of a partition of $\mathbb N$, J. Combin. Theory (Ser. A) 48 (1988), 39--52. MR 89m:04003
  • 5. ------, Nonmetrizable topological dynamics and Ramsey theory, Trans. Amer. Math. Soc. 320 (1990), 293--320. MR 90h:03046
  • 6. ------, On $IP^*$ sets and central sets, Combinatorica 14 (1994), 269--277. CMP 95:05
  • 7. J. Berglund, H. Junghenn, and P. Milnes, Analysis on semigroups, Wiley, New York, 1989. MR 91b:43001
  • 8. M. Boshernitzan and A. Fraenkel, Nonhomogeneous spectra of numbers, Discrete Math. 34 (1981), 325--327. MR 82d:10077
  • 9. R. Ellis, A semigroup associated with a transformation group, Trans. Amer. Math. Soc. 94 (1960), 272--281. MR 23:A961
  • 10. ------, Lectures on topological dynamics, New York, Benjamin, 1969. MR 42:2463
  • 11. A. Fraenkel, Complementary systems of integers, Amer. Math. Monthly 84 (1977), 114--115. MR 55:2825
  • 12. H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton Univ. Press, Princeton, NJ, 1981. MR 82j:28010
  • 13. H. Furstenberg and B. Weiss, Simultaneous diophantine approximation and $IP$-sets, Acta Arith. 49 (1988), 413--426. MR 89f:11097
  • 14. ------, Topological dynamics and combinatorial number theory, J. Analyse Math. 34 (1978), 61--85. MR 80g:05009
  • 15. R. Graham, On a theorem of Uspensky, Amer. Math. Monthly 70 (1963), 407--409. MR 26:6062
  • 16. R. Graham, S. Lin, and C. Lin, Spectra of numbers, Math. Magazine 51 (1978), 174--176. MR 58:10808
  • 17. G. Hardy and E. Wright, An introduction to the theory of numbers, Oxford Univ. Press, Oxford, 1979. MR 81i:10002
  • 18. N. Hindman, Finite sums from sequences within cells of a partition of $\mathbb N$, J. Combin. Theory (Ser. A) 17 (1974), 1--11. MR 50:2067
  • 19. ------, Partitions and sums and products of integers, Trans. Amer. Math. Soc. 247 (1979), 19--32. MR 80b:10022
  • 20. ------, Summable ultrafilters and finite sums, Logic and Combinatorics, (S. Simpson, ed.), Contemporary Math., vol. 65, Amer. Math. Soc. Providence, RI, 1987, pp. 263--274. MR 88h:03070
  • 21. ------, The existence of certain ultrafilters on $\mathbb N$ and a conjecture of Graham and Rothschild, Proc. Amer. Math. Soc. 36 (1972), 341--346. MR 46:7041
  • 22. ------, Ultrafilters and combinatorial number theory, Number Theory Carbondale 1979, (M. Nathanson, ed.), Lecture Notes in Math., vol. 751, Springer, 1979, pp. 119--184. MR 81m:10019
  • 23. N. Hindman and J. Pym, Free groups and semigroups in $\beta\mathbb N$, Semigroup Forum 30 (1984), 177--193. MR 86c:22002
  • 24. J. Kelley, General topology, Van Nostrand, New York, 1955. MR 16:1136c
  • 25. B. Kra, A dynamical approach to central sets and iterated spectra of numbers, Abstracts Amer. Math. Soc. 13 (1992), 294.
  • 26. J. Lawson and A. Lisan, Transitive flows: a semigroup approach, Mathematika 38 (1991), 348--361. MR 93c:22003
  • 27. I. Niven, Diophantine approximations, Interscience, New York, 1963. MR 26:6120
  • 28. I. Schoenberg, Mathematical time exposures, Mathematical Association of America, Washington, DC, 1982. MR 85b:00001
  • 29. T. Skolem, On certain distributions of integers in pairs with given differences, Math. Scand. 5 (1957), 57--68. MR 19:1159g
  • 30. ------, Über einige Eigenschaften der Zahlenmengen $[\alpha n+\beta]$ bei irrationalem $\alpha$ mit einleitenden Bemerkungen über dinige kombinatorische probleme, Norske Vid. Selsk. Forh. 30 (1957), 42--49. MR 19:1159i
  • 31. J. Strutt (Lord Rayleigh), The theory of sound, Macmillan, London, 1977; Reprinted, Dover, New York, 1945.
  • 32. J. Uspensky, On a problem arising out of a certain game, Amer. Math. Monthly 34 (1927), 516--521.
  • 33. B. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 19 (1927), 212--216.

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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: https://doi.org/10.1090/S0002-9947-96-01533-4
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

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