From minimal-length quantum theory to modified gravity

In this work, we consider generalized uncertainty principles (GUPs) that incorporate a minimal length through generic momentum-dependent deformation functions. We thus develop a systematic approach connecting such a framework to effective gravitational actions extending general relativity. By examining quantum gravity-motivated corrections to black hole entropy induced by the GUP and employing Wald's formalism, we reconstruct modifications to Einstein's gravity within the contexts of $f(R)$ and $f(R, R_{\mu\nu} R^{\mu\nu})$ theories. In this way, we establish a direct mapping between the GUP parameters and the higher-order curvature coefficients in the gravitational Lagrangian. As an illustrative application, we compute corrections to the general relativistic prediction for light deflection, which in turn allows us to infer a stringent upper bound on the minimal measurable length. Our results show that GUP-induced effects can be consistently embedded into extended gravity theories, offering a promising framework for testing quantum gravity phenomenology through astrophysical and cosmological observations.