Distality for the Asymptotic Couple of the Field of Logarithmic Transseries

Distality in valued fields and related structures

“…It is well-known that if T is distal, then T is NIP; for instance, see [34,Proposition 2.8]. Distality can be thought of as a notion of pure instability among NIP theories.…”

Hamel Spaces and Distal Expansions

“…Proof. It is well known that distality implies NIP; e.g., see [7,Proposition 2.8] for a proof. Proof.…”

“…Proof. This is an exercise in simplification and bookkeeping which mimics the proof of [7,Lemma 4.4], except that it uses the fact that the value set v(G = ) is a C -linearly independent subset of G (Lemma 4.3).…”

“…This is routine and left to the reader. See the proof of[7, Lemma 4.3]. Let h(x, y) be an L-term of arity m + n, b ∈ M n , and (a i ) i∈I an indiscernible sequence from v(M) m , with a i…”

Distality Rank

“…Hieronymi and Nell developed criteria for determining when certain expansions of o-minimal theories remain distal [13], and Nell continued this work by studying distal behavior in dense pairs of o-minimal structures [18]. In 2018, the asymptotic couple of the field of logarithmic transseries was shown to be distal by Gehret and Kaplan [10], and in 2020, Aschenbrenner, Chernikov, Gehret, and Ziegler explored distality in valued fields and, among other things, proved that the differential field of the logarithmic-exponential transseries is distal [1].…”