Born-Infeld-Hořava gravity

We define various Born-Infeld gravity theories in $3+1$ dimensions which reduce to Hořava’s model at the quadratic level in small curvature expansion. In their exact forms, our actions provide $z\to \infty$ extensions of Hořava’s gravity, but when small curvature expansion is used, they reproduce finite $z$ models, including some half-integer ones.