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Small sets containing any pattern

Published online by Cambridge University Press:  31 July 2018

URSULA MOLTER  and
ALEXIA YAVICOLI
URSULA MOLTER
Affiliation:
Departamento de Matemática and IMAS/UBA-CONICET, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pab. I, (1428) CABA, Argentina. e-mail: umolter@dm.uba.ar; ayavicoli@dm.uba.ar
ALEXIA YAVICOLI
Affiliation:
Departamento de Matemática and IMAS/UBA-CONICET, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pab. I, (1428) CABA, Argentina. e-mail: umolter@dm.uba.ar; ayavicoli@dm.uba.ar
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Abstract

Given any dimension function h, we construct a perfect set E ⊆ ${\mathbb{R}}$ of zero h-Hausdorff measure, that contains any finite polynomial pattern.

This is achieved as a special case of a more general construction in which we have a family of functions $\mathcal{F}$ that satisfy certain conditions and we construct a perfect set E in ${\mathbb{R}}^N$, of h-Hausdorff measure zero, such that for any finite set {f1,. . .,fn} ⊆ $\mathcal{F}$, E satisfies that $\bigcap_{i=1}^n f^{-1}_i(E)\neq\emptyset$.

We also obtain an analogous result for the images of functions. Additionally we prove some related results for countable (not necessarily finite) intersections, obtaining, instead of a perfect set, an $\mathcal{F}_{\sigma}$ set without isolated points.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 1 , January 2020 , pp. 57 - 73
Copyright
Copyright © Cambridge Philosophical Society 2018 

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Footnotes

†The research for this paper was partially supported by grants UBACyT 2014-2017 20020130100403BA, PIP 11220110101018 (CONICET) and PICT 2014 - 1480, MinCyT.

References

REFERENCES

[1] Bergelson, V. and Leibman, A. Polynomial extensions of van der Waerden's and Szemerédi's theorems. J. Amer. Math. Soc. 9 (3)(1996), 725–753.Google Scholar
[2] Boshernitzan, M. and Chaika, J. Dichotomy for Arithmetic Progressions in subsets of reals. Proc. Amer. Math. Soc. 144 (12) (2016), 5029–5034.Google Scholar
[3] Chan, V., Łaba, I. and Pramanik, M. Finite configurations in sparse sets. J. Anal. Math. 128 (2016), 289–335.Google Scholar
[4] Darji, U.B. and Keleti, T. Covering ${\mathbb{R}}$ with translates of a compact set. Proc. Amer. Math. Soc. 131 (8) (2003), 2598–2596Google Scholar
[5] Davies, R.O., Marstrand, J.M. and Taylor, S.J. On the intersections of transforms of linear sets. Colloq. Math. 7 (1959/1960), 237–243.Google Scholar
[6] Erdős, P. Remarks on some problems in number theory Math. Balkanica 4 (1974), 197–202.Google Scholar
[7] Falconer, K. J. On a problem of Erdős on sequences and measurable sets. Proc. Amer. Math. Soc 90 (1984), 77–78.Google Scholar
[8] Falconer, K. J. On a problem of Erdős on fractal combinatorial geometry. J. Combin. Theory Ser. A 59 (1) (1992), 142–148.Google Scholar
[9] Fraser, R. and Pramanik, M. Large sets avoiding patterns Anal. PDE 11 (5) (2018), 1083–1011.Google Scholar
[10] Furstenberg, H.. Recurrence in ergodic theory and combinatorial number theory. (Princeton University Press, Princeton, N.J., m. B. Porter Lectures, 1981).Google Scholar
[11] Henriot, K., Łaba, I. and Pramanik, M. On polynomial configurations in fractal sets. Anal. PDE 9 (5) (2016), 1153–1184.Google Scholar
[12] Iosevich, A. and Liu, B.. Equilateral triangles in subsets of R d of large hausdorff dimension. Preprint, arXiv:1603.01907 (2016).Google Scholar
[13] Łaba, I. and Pramanik, M. Arithmetic progressions in sets of fractional dimension. Geom. Funct. Anal. 19 (2009), 429–456.Google Scholar
[14] Keleti, T. Construction of one-dimensional subsets of the reals not containing similar copies of given patterns. Anal. PDE 1 (1) (2008), 29–33.Google Scholar
[15] Keleti, T., Nagy, D.T. and Shmerkin, P. Squares and their centers. J. Anal. Math. 134 (2) (2014), 643–669.Google Scholar
[16] Maga, P. Full dimensional sets without given patterns. Real Anal. Exchange 36 (1) (2010/2011), 79–90.Google Scholar
[17] Máthé, A. Sets of large dimension not containing polynomial configurations. Avd. Math. 316 (2017), 691–709.Google Scholar
[18] Szemerédi, E. On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27 (1975), 199–245, collection of articles in memory of Juriĭ Vladimirovič Linnik.Google Scholar
[19] Tao, T. and Ziegler, T. The primes contain arbitrarily long polynomial progressions. Acta Math. 201 (2008), 213–305.Google Scholar
[20] Tao, T. and Ziegler, T. Polynomial Patterns in the Primes. Forum Math. Pi 6 (e1) (2018), 60pp.Google Scholar
[21] Yavicoli, A. Large sets avoiding linear patterns. Proc. Amer. Math. Soc. To appear (2017).Google Scholar