Abstract
We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. LetE be a measurable subset of ℝm, with\(\bar D(E) > 0\). LetV = {0,v 1,...,v k} ⊂ ℝm. We show that fort large enough, we can find an isometric copy oftV arbitrarily close toE. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss [FuKaW] showing a similar property form=k=2.
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References
[Bo] J. Bourgain,A Szemerédi type theorem for sets of positive density in ℝ k, Israel J. Math.54 (1986), 307–316.
[FM] K. J. Falconer and J. M. Marstrand,Plane sets with positive density at infinity contain all large distances, Bull. Lond. Math. Soc.18 (1986), 471–474.
[Fu1] H. Furstenberg,Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, NJ, USA, 1981.
[Fu2] H. Furstenberg,Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math.31 (1977), 204–256.
[FrKr] N. Frantzikinakis and B. Kra,Convergence of multiple ergodic averages for some commuting transformations, Ergodic Theory Dynam. Systems,25 (2005), 799–809.
[FuKa] H. Furstenberg, and Y. Katznelson,An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math.34 (1978), 275–291.
[FuKaW] H. Furstenberg, Y. Katznelson and B. Weiss,Ergodic theory and configurations in sets of positive density, Mathematics of Ramsey Theory, Algorithms Combin., 5, Springer, Berlin, 1990, pp. 184–198.
[Gr] R. L. Graham,Recent trends in Euclidean Ramsey theory, Discrete Math.136 (1994), 119–127.
[HKr] B. Host and B. Kra,Non-conventional ergodic averages and nilmanifolds, Annals of Math. (2)161 (2005), 397–488.
[Le] A. Leibman,Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems25 (2005), 201–213.
[Le1] A. Leibman,Personal communication.
[Le2] A. Leibman,Host-Kra and Ziegler factors and convergence of multiple averages, Handbook of Dynamical Systems, vol. 1B, B. Hasselblatt and A. Katok, eds., Elsevier, 2005, pp. 841–853.
[Ma] A. Malcev,On a class of homogeneous spaces, Izvestiya Akad. Nauk SSSR, Ser. Mat.13 (1949), 9–32.
[Pa1] W. Parry,Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math.91 (1969), 757–771.
[Pa2] W. Parry,Metric classification of ergodic nilflows and unipotent affines, Amer. J. Math.93 (1969), 819–828.
[Pe] K. Petersen,Ergodic theory, Cambridge University Press, 1983.
[PS] C. Pugh and M. Shub,Ergodic elements of ergodic actions, Compositio Mathematica23 (1971), 115–122.
[Sh] N. Shah,Invariant measures and orbit closures on homogeneous spaces for actions of subgroups, Lie Groups and Ergodic Theory (Mumbai, 1996), 229–271, Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998.
[Z] T. Ziegler,Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc., posted on March 17, 2006, PII: S 0894-0347(06)00532-7 (to appear).
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Ziegler, T. Nilfactors of ℝm and configurations in sets of positive upper density in ℝm . J. Anal. Math. 99, 249–266 (2006). https://doi.org/10.1007/BF02789447
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DOI: https://doi.org/10.1007/BF02789447