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Product of Simplices and sets of positive upper density in ℝd

  • NEIL LYALL (a1) and ÁKOS MAGYAR (a1)
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 dk1 + 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 dk + 1, a result originally due to Bourgain, is also presented.

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References
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[1] Bourgain, J. A Szemerédi type theorem for sets of positive density in Rk. Israel J. Math. 54 (1986), no. 3, 307316.
[2] Furstenberg, H., Katznelson, Y. and Weiss, B. Ergodic theory and configurations in sets of positive density. Israel J. Math. 54 (1986), no. 3, 307316.
[3] Stein, E. Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals (Princeton University Press, Princeton, NJ., 1993).
[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. 84112.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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