Translation properties of sets of positive upper density
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- by Vitaly Bergelson and Benjamin Weiss PDF
- Proc. Amer. Math. Soc. 94 (1985), 371-376 Request permission
Abstract:
Generalizing a result of Raimi we show that there exists a set $E \subset {\mathbf {N}}$ such that if $A \subset {\mathbf {N}}$ is a set with positive upper density, then there exists a number $k \in {\mathbf {N}}$ such that ${d^ * }((A + k) \cap E) > 0$ and ${d^ * }((A + k) \cap {E^c}) > 0$. Some extensions and further results are also obtained.References
- Ralph A. Raimi, Translation properties of finite partitions of the positive integers, Fund. Math. 61 (1967/68), 253–256. MR 222874, DOI 10.4064/fm-61-3-253-256
- C. Ryll-Nardzewski, Remark on Raimi’s theorem on translations, Fund. Math. 61 (1967/68), 257–258. MR 222875, DOI 10.4064/fm-61-3-257-258
- Neil Hindman, Ultrafilters and combinatorial number theory, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 119–184. MR 564927 D. G. Champernowne, The construction of decimals normal in the scale of ten, J. London Math. Soc. 8 (1933), 254-260.
- R. v. Mises, Über Zahlenfolgen, die ein kollektiv-ähnliches Verhalten zeigen, Math. Ann. 108 (1933), no. 1, 757–772 (German). MR 1512874, DOI 10.1007/BF01452862
- H. Davenport and P. Erdös, Note on normal decimals, Canad. J. Math. 4 (1952), 58–63. MR 47084, DOI 10.4153/cjm-1952-005-3
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 94 (1985), 371-376
- MSC: Primary 11K16
- DOI: https://doi.org/10.1090/S0002-9939-1985-0787875-3
- MathSciNet review: 787875