Born-Infeld-Hořava gravity

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