Correction to: On groups interpretable in various valued fields (Selecta Mathematica, (2024), 30, 4, (59), 10.1007/s00029-024-00946-2)

This correction concerns [1]. Remark 4.19 (2) of [1] states that: If D is a vicinic set and S is a definable set of finite dp-rank then if X₁,X₂⊆S are (almost) D-sets then so is X₁×X₂⊆S².The proof provided for this statement is wrong, and we do not know whether the statement is true in general vicinic sets (though we can prove it for SW-uniformities). The error in the proof is, in the language of [1], that we do not know whether, for a definable set S, the (almost) D-critical rank of S×S is twice the (almost) D-critical rank of S. Below, we explain how Remark 4.19 (2) can be avoided in our arguments. If D is a vicinic set and S is a definable set of finite dp-rank then if X₁,X₂⊆S are (almost) D-sets then so is X₁×X₂⊆S². Throughout [1] Remark 4.19 (2) is only applied for D-sets (and not for almost D-sets), so we recall the definition of the former. Let D and S be definable sets of finite dp-rank. An infinite definable set X⊆S is a D-set if there exists a definable injection f:X→Dⁿ (some n), X is D-critical for S, i.e., has maximal dp-rank among all definable subsets of S satisfying (1) above, and f(X) has minimal fibers. there exists a definable injection f:X→Dⁿ (some n), X is D-critical for S, i.e., has maximal dp-rank among all definable subsets of S satisfying (1) above, and f(X) has minimal fibers. Since the notion of having minimal fibres is inconsequential for this correction, we will not dwell on it. Remark 4.19 (2) of [1] was used explicitly only in Lemma 5.7, on which also the proof of Lemma 5.10 depends. These lemmas are stated for D-sets. Below we show that this assumption can be relaxed, so that Remark 4.19(2) need not be invoked. Lemma 5.10 and lemmas 5.5-5.8 on which it depends, all relate to infinitesimal vicinities (Definition 5.4) that were only introduced in the context of D-sets. We will show that this definition can be extended to a larger class of sets, and that the proofs of the above-mentioned lemmas extend to this wider context unaltered. The following ad hoc definition is all we need to extend the scope of our infinitesimal vicinities. It merely drops clause (2) in Definition 0.1 above: An infinite definable set X⊆S is a semi-D-set over A, if there exists an A-definable injection f:X→Dⁿ, such that f(X)⊆Dⁿ has minimal fibres. The definition of infinitesimal vicinities extends automatically from the class of D-sets to the class of semi-D-sets: Let Z⊆G be a semi-D-set over A and d∈Z an A-generic point. The infinitesimal vicinity of d in Z, denoted ν_Z(d), is the partial type consisting of all B-generic vicinities of d in Z, as B varies over all small parameter subsets of M. We now note that the proofs of lemmas 5.5-5.8 go through unaltered to this wider class of infinitesimal vicinities. I.e., they remain true for semi-D-sets (dropping the D-criticality assumption). Indeed, the only lemma on which these results rely, and where D-sets are mentioned, is Lemma 4.20. Lemma 4.20, in turn, only requires D-sets in order to invoke Remark 4.19(1). The proof of Remark 4.19(1) extends unaltered to semi-D-set, as it does not use the D-criticality of D-sets. Thus, Lemma 5.5 through Lemma 5.8 remain true, as stated, and without any changes to their proofs, in the more general setting of semi-D-sets and their associated infinitesimal vicinities, as introduced above. Finally, and crucially, we note that the proof of Proposition 5.10 is still valid, despite the explicit assumption that the sets concerned areD-sets. In the 14-th line of the proof (in the reference to Lemma 5.8), we consider the map (x,y)↦x·y and then apply Lemma 5.8 to show that it maps ν_X(c)×ν_Y(d) into ν_Z(cd) (here X, Y, Z are D-sets). This result implicitly used the fact that ν_X(c)×ν_Y(d)=ν_X×Y(c,d), as follows from Lemma 5.7. While this was wrong in our original definition of infinitesimal vicinities (we do not know that X×Y is D-critical so ν_X×Y(c,d) is not defined), X×Y is a semi-D-set so the generalized version of Lemma 5.7 (and consequently Lemma 5.8) may be applied. We also note that having withdrawn Remark 4.19 (2), we do not know whether ν_D(G×G)=ν_D(G)×ν_D(G) for general vicinic D. This question, while of possible interest, does not affect any of the results of [1].