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

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

2007-09-27

arxiv.org/abs/0709.4432v1

Mathematics

Number Theory

12 pages

Scientific paper

Let a be a real number between 0 and 1. Ernie Croot showed that the quantity \max_A #(3-term arithmetic progressions in A)/p^2, where A ranges over all subsets of Z/pZ of size at most a*p, tends to a limit as p tends to infinity through primes. Writing c(a) for this limit, we show that c(a) = a^2/2 provided that a 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/pZ of cardinality m, provided that m < c*p.

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