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Product of Simplices and sets of positive upper density in ℝd

Published online by Cambridge University Press:  02 May 2017

NEIL LYALL  and
ÁKOS MAGYAR
NEIL LYALL
Affiliation:
Department of Mathematics, The University of Georgia, Athens, GA 30602, U.S.A. e-mail: lyall@math.uga.edu; magyar@math.uga.edu
ÁKOS MAGYAR
Affiliation:
Department of Mathematics, The University of Georgia, Athens, GA 30602, U.S.A. e-mail: lyall@math.uga.edu; magyar@math.uga.edu
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Abstract

We establish that any subset of ℝd of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided d ⩾ 4.

We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if Δk1 and Δk2 are two fixed non-degenerate simplices of k1 + 1 and k2 + 1 points respectively, then any subset of ℝd of positive upper Banach density with d ⩾ k1 + k2 + 6 will necessarily contain an isometric copy of all sufficiently large dilates of Δk1 × Δk2.

A new direct proof of the fact that any subset of ℝd of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of k + 1 points provided d ⩾ k + 1, a result originally due to Bourgain, is also presented.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 1 , July 2018 , pp. 25 - 51
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Bourgain, J. A Szemerédi type theorem for sets of positive density in Rk. Israel J. Math. 54 (1986), no. 3, 307–316.Google Scholar
[2] Furstenberg, H., Katznelson, Y. and Weiss, B. Ergodic theory and configurations in sets of positive density. Israel J. Math. 54 (1986), no. 3, 307–316.Google Scholar
[3] Stein, E. Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals (Princeton University Press, Princeton, NJ., 1993).Google Scholar
[4] Komlós, J., Shokoufandeh, A., Simonovits, M. and Szemerédi, M. E. The regularity lemma and its applications in graph theory. In Theoretical Aspects of Computer Science (Springer Berlin Heidelberg., 2002), pp. 84–112.Google Scholar