An effective version of Katok's horseshoe theorem for conservative $C^2$ surface diffeomorphisms

For area preserving $C^2$ surface diffeomorphisms, we give an explicit finite information condition, on the exponential growth of the number of Bowen’s $(n,\delta )-$balls needed to cover a positive proportion of the space, that is sufficient to guarantee positive topological entropy. This can be seen as an effective version of Katok’s horseshoe theorem in the conservative setting. We also show that the analogous result is false in dimension larger than $3$.