On Arithmetic Progressions in Symmetric Sets in Finite Field Model - Hązła, Jan (The Electronic Journal of Combinatorics, 2020)
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 Z_q^n containing |S| = μ · q^n elements must contain at least δ(q, μ) · 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 SF_p^n with |S|=μ· p^n elements contains at least μ^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.