Dimension theory for the asymptotic couple of the field of logarithmic transseries

In this paper we completely characterize all dimension functions on all models of the theory $T_{\log }$ of the asymptotic couple of the field of logarithmic transseries (Dimension Theorem). This is done by characterizing the “small” $1$-variable definable sets (Small Sets Theorem). As a byproduct, we show that $T_{\log }$ is d-minimal and does not eliminate imaginaries. Separately, we provide an abstract criterion for d-minimality, which we use to observe some new examples of d-minimal expansions of valued fields.