Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T18:23:23.371Z Has data issue: false hasContentIssue false
  • English
  • Français

Structure Along Arithmetic Patterns in Sequences of Vectors

Published online by Cambridge University Press:  30 March 2009

P. CANDELA
P. CANDELA*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss a new direction in which the use of some methods from arithmetic combinatorics can be extended. We consider functions taking values in Euclidean space and supported on subsets of {1, 2, . . ., N}. In this context we present a proof of a natural generalization of Szemerédi's theorem. We also prove a similar generalization of a theorem of Sárkőzy using a vector-valued Fourier transform, adapting an argument of Green and obtaining effective bounds.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 6 , November 2009 , pp. 861 - 870
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bollobás, B. (1998) Modern Graph Theory, Vol. 184 of Graduate Texts in Mathematics, Springer.CrossRefGoogle Scholar
[2]Gowers, W. T. (2001) A new proof of Szemerédi's theorem. Geom. Funct. Anal. 11 465–588.CrossRefGoogle Scholar
[3]Green, B. J. (2002) On arithmetic structures in dense sets of integers. Duke Math. J. 114 215–238.CrossRefGoogle Scholar
[4]Gunderson, D. S., Leader, I., Prömel, H. J. and Rödl, V. (2001) Independent arithmetic progressions in clique-free graphs on the natural numbers. J. Combin. Theory Ser. A 93 1–17.CrossRefGoogle Scholar
[5]Sárkőzy, A. (1978) On difference sets of sequences of integers I. Acta. Math. Sci. Hungar. 31 125–149.CrossRefGoogle Scholar
[6]Simonovits, M. and Sós, V. (2001) Ramsey–Turan theory. Discrete Math. 229 293–340.CrossRefGoogle Scholar
[7]Varnavides, P. (1959) On certain sets of positive density. J. London Math. Soc. 34 358–360.CrossRefGoogle Scholar