In this paper we explore some properties of H-structures which are introduced in [2].We describe a construction of H-structures based on one-dimensional asymptotic classes which preserves pseudo-finiteness. That is, the H-structures we construct are ultraproducts of finite structures.We also prove that under the assumption that the base theory is supersimple of SU -rank one, there are no new definable groups in H-structures. This improves the corresponding result in [2]. 1 3. (Density/coheir property) If A ⊆ M is finite dimensional and q ∈ S 1 (A) is nonalgebraic, there is a ∈ H(M ) such that a |= q; (Extension property) IfA ⊆ M is finite dimensional and q ∈ S 1 (A) is nonalgebraic, then there is a ∈ M , a |= q and a ∈ acl L (A ∪ H(M )).Equivalently, we can replace density and extension properties with the following more general ones:• (Generalised density/coheir property) If A ⊆ M is finite dimensional and q ∈ S n (A) has dimension n, then there is a ∈ H(M ) n such that a |= q;• (Generalised extension property) If A ⊆ M is finite dimensional and q ∈ S n (A) is non-algebraic, then there is a ∈ M n , a |= q andA structure M is called an H-structure if it is an H-expansion of some model of a geometric theory.H-structures are closely related to lovely pairs, where, instead of an independent subset, a dense and co-dense elementary substructure is added. We recall the definition of lovely pairs in the special case that the base theory is geometric, see [1].Definition 2. Let T be a geometric theory in a language L and let L P be the expansion of L by a unary predicate P . An L P -structure (M, N ) is a lovely pair of models of T , if 1. M |= T ; 2. N is an L-elementary submodel of M ; 3. (Density/coheir property) If A ⊆ M is finite dimensional and q ∈ S 1 (A) is nonalgebraic, there is a ∈ N such that a |= q; 4. (Extension property) If A ⊆ M is finite dimensional and q ∈ S 1 (A) is nonalgebraic, then there is a ∈ M , a |= q and a ∈ acl L (A ∪ N ).Fact 3. [2], [1]. Properties of H-structures and lovely pairs. Let T be a complete geometric theory in a language L. • H-expansions of models of T exist and all of them are L H -elementary equivalent. Let T H be the corresponding theory. Similarly, lovely pairs of models of T exist, and all of them are L P -elementary equivalent.• If the geometry of T is nontrivial and T is strongly minimal/supersimple/superrosy of rank 1, then T H is ω-stable/supersimple/superrosy of rank ω.• Let (M, H(M )) be an H-structure. Then (M, acl L (H(M ))) is a lovely pair.Consider the theory of pseudofinite fields. It is supersimple of SU -rank one. By the fact above, H-expansions and lovely pairs of pseudofinite fields exist. However, the proof of existence uses general model theoretic techniques such as saturated models and union of chains. It is not clear whether it is possible to have H-expansions or lovely pairs of pseudofinite fields that are ultraproducts of finite structures.The answer turns out to be negative for lovely pairs.

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integervalued. Moreover, we establish a partial connection between coarse dimension and transformal transcendence degree in these difference fields.

We prove Schlichting's theorem for approximate subgroups: if X is a uniform family of commensurable approximate subgroups in some ambient group, then there exists an invariant approximate subgroup commensurable with X .

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.