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We consider two problems regarding arithmetic progressions in symmetric sets in the finite field (product space) model. First, we show that a symmetric set S subset of Z(q)(n) containing vertical bar S vertical bar = mu . q(n) elements must contain at least delta(q, mu) . q(n) . 2(n) arithmetic progressions x, x+d, . . . , x+(q - 1).d such that the difference d is restricted to lie in {0, 1}(n). Second, we show that for prime p a symmetric set S subset of F-p(n) with vertical bar S vertical bar = mu . p(n) elements contains at least mu(C(p)) . p(2n) arithmetic progressions of length p. This establishes that the qualitative behavior of longer arithmetic progressions in symmetric sets is the same as for progressions of length three.
Thomas Mountford, Michael Cranston
Donna Testerman, Martin W. Liebeck