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Bulletin of the London Mathematical Society Advance Access originally published online on August 20, 2008
Bulletin of the London Mathematical Society 2008 40(6):945-955; doi:10.1112/blms/bdn074
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© 2008 London Mathematical Society

On the maximal number of 3-term arithmetic progressions in subsets of Z/pZ

Ben Green and Olof Sisask

Centre for Mathematical Sciences
Wilberforce Road
Cambridge
CB3 0WB
United Kingdom
b.j.green@dpmms.cam.ac.uk

Received 27 September 2007. Revision received 20 May 2008.

Let {alpha} isin [0, 1] be a real number. Ernie Croot (Canad. Math. Bull. 51 (2008) 47–56) showed that the quantity maxA # (3-term arithmetic progressions in A)/p2, where A ranges over all subsets of Z/pZ of size at most {alpha} p, tends to a limit as p -> {infty} through primes. Writing c({alpha}) for this limit, we show that c({alpha})={alpha}2/2 provided that {alpha} is smaller than some absolute constant. In fact, we prove rather more, establishing a structure theorem for sets having the maximal number of 3-term progressions amongst all subsets of Z/p Z of cardinality m, provided that m < cp.

2000 Mathematics Subject Classification 11B99.

The first author was a Clay Research Fellow while this work was carried out and gratefully acknowledges the support of the Clay Institute. The second author was funded by an EPSRC DTG through the University of Bristol. While this work was being carried out, he spent time at MIT and the University of Cambridge, and would like to thank both the institutions for their kind hospitality.


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