Weyl gauge invariant DBI action in conformal geometry

We construct the analogue of the Dirac-Born-Infeld (DBI) action in Weyl conformal geometry in $d$ dimensions and obtain a general theory of gravity with Weyl gauge symmetry of dilatations (Weyl-DBI). This is done in the Weyl gauge covariant formulation of conformal geometry in $d$ dimensions, suitable for a gauge theory, in which this geometry is metric. The Weyl-DBI action is a special gauge theory in that it has the same gauge invariant expression with dimensionless couplings in any dimension $d$, with no need for a UV regulator (be it a DR subtraction scale, field or higher derivative operator) for which reason we argue it is Weyl-anomaly free. For $d=4$ dimensions, the leading order of a series expansion of the Weyl-DBI action recovers the gauge invariant Weyl quadratic gravity action associated to this geometry, that is Weyl anomaly-free; this is broken spontaneously and Einstein-Hilbert gravity is recovered in the broken phase, with $\Lambda>0$. All the remaining terms of this series expansion are of non-perturbative nature but can, in principle, be recovered by (perturbative) quantum corrections in Weyl quadratic gravity in $d=4$ in a gauge invariant (geometric) regularisation, provided by the Weyl-DBI action. If the Weyl gauge boson is not dynamical the Weyl-DBI action recovers in the leading order the conformal gravity action. All fields and scales have geometric origin, with no added matter, scalar field compensators or UV regulators.