 subgroup growth) and number theory by various authors, notably Dan Segal and Marcus du Sautoy.Macintyre worked with Zoé Chatzidakis and Lou van den Dries on definable sets over finite fields generalising the estimates of Serge Lang and André Weil to definable sets an revisiting the work of James Ax on the logic of finite and pseudofinite fields.He initiated and proved results on 
the model theory of difference fields and of Frobenius automorphisms, where he proved extensions of Ax's work to this setting (including model-companions and decidability).