Definable sets in automorphism groups of rational order

The main result of the paper is describing all definable subsets in the group $\operatorname{Aut}\langle\mathbb Q,\le\rangle$ of all automorphisms of the natural ordering on the rational numbers, and also in groups of the form $\operatorname{Aut}_\mathbf I\langle\mathbb Q,\le\rangle$, where $\mathbf I$ is a Turing ideal consisting of elements of $\operatorname{Aut}\langle\mathbb Q,\le\rangle$ whose Turing degree is contained in $\mathbf I$. This description is properly a uniform method for proving definability of all basic properties appearing in works on the theory of groups $\operatorname{Aut}_\mathbf I\langle\mathbb Q,\le\rangle$, as well as definability of a number of new sets. Also, we describe automorphism groups for such groups $\operatorname{Aut}_\mathbf I\langle\mathbb Q,\le\rangle$ and state a number of structure properties for elementary subgroups in $\operatorname{Aut}\langle\mathbb Q,\le\rangle$.