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Simplices and sets of positive upper density in $ \mathbb{R}^d$


Authors: Lauren Huckaba, Neil Lyall and Ákos Magyar
Journal: Proc. Amer. Math. Soc. 145 (2017), 2335-2347
MSC (2010): Primary 11B30, 42B25, 42A38
DOI: https://doi.org/10.1090/proc/13538
Published electronically: January 25, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove an extension of Bourgain's theorem on pinned distances in a measurable subset of $ \mathbb{R}^2$ of positive upper density, namely Theorem $ 1^\prime $ in a 1986 article, to pinned non-degenerate $ k$-dimensional simplices in a measurable subset of $ \mathbb{R}^{d}$ of positive upper density whenever $ d\geq k+2$ and $ k$ is any positive integer.


References [Enhancements On Off] (What's this?)

  • [1] J. Bourgain, A Szemerédi type theorem for sets of positive density in $ {\bf R}^k$, Israel J. Math. 54 (1986), no. 3, 307-316. MR 853455, https://doi.org/10.1007/BF02764959
  • [2] J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69-85. MR 874045, https://doi.org/10.1007/BF02792533
  • [3] Hillel Furstenberg, Yitzchak Katznelson, and Benjamin Weiss, Ergodic theory and configurations in sets of positive density, Mathematics of Ramsey theory, Algorithms Combin., vol. 5, Springer, Berlin, 1990, pp. 184-198. MR 1083601, https://doi.org/10.1007/978-3-642-72905-8_13
  • [4] Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437
  • [5] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192

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Additional Information

Lauren Huckaba
Affiliation: Department of Mathematics, The University of Georgia, Athens, Georgia 30602
Email: lhuckaba@math.uga.edu

Neil Lyall
Affiliation: Department of Mathematics, The University of Georgia, Athens, Georgia 30602
Email: lyall@math.uga.edu

Ákos Magyar
Affiliation: Department of Mathematics, The University of Georgia, Athens, Georgia 30602
Email: magyar@math.uga.edu

DOI: https://doi.org/10.1090/proc/13538
Received by editor(s): April 1, 2016
Received by editor(s) in revised form: July 21, 2016
Published electronically: January 25, 2017
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2017 American Mathematical Society

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