Adapting the slice rank method yields that sets in F_q^n without 3-APs with differences in S^n have size at most q^{(1-ε_q)n} when |S|>(q+1)/2 and q is an odd prime power.
Tao, A symmetric formulation of the Croot–Lev–Pach–Ellenberg–Gijswijt capset bound, blog post, 2016, http://terrytao.wordpress.com/2016/05/18/
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A note on arithmetic progressions with restricted differences
Adapting the slice rank method yields that sets in F_q^n without 3-APs with differences in S^n have size at most q^{(1-ε_q)n} when |S|>(q+1)/2 and q is an odd prime power.