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On the Distribution of Three-Term Arithmetic Progressions in Sparse Subsets of Fpn

Published online by Cambridge University Press:  18 August 2011

HOI H. NGUYEN
HOI H. NGUYEN*
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA (e-mail: hoi@math.rutgers.edu)
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Abstract

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We give a short proof of the following result on the distribution of three-term arithmetic progressions in sparse subsets of Fpn. For every α > 0 there exists a constant C = C(α) such that the following holds for all r ≥ Cpn/2 and for almost all sets R of size r of Fpn. Let A be any subset of R of size at least αr; then A contains a non-trivial three-term arithmetic progression. This is an analogue of a hard theorem by Kohayakawa, Łuczak and Rödl. The proof uses a version of Green's regularity lemma for subsets of a typical random set, which is of interest in its own right.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 5 , September 2011, pp. 777 - 791
Copyright
Copyright © Cambridge University Press 2011

References

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