Searching for GEMS: TOI-5916 b & TOI-6158 b are two Saturn-density planets orbiting M2 dwarfs

We confirm the planetary nature of (1) TOI-5916 b and (2) TOI-6158 b, two Exoplanets Transiting M-dwarf Stars (GEMS), both discovered by the Transiting Exoplanet Survey Satellite (TESS). Both systems were confirmed with ground-based photometry (Red Buttes Observatory and Swope, respectively) and radial velocity data from the Habitable-zone Planet Finder. Their radii are $R_1=11.8_{-0.51}^{+0.52}{R}_{\oplus }$ and $R_2=10.4_{-1.11}^{+2.70}{R}_{\oplus }$ and masses are $M_1=219\pm 28M_{\oplus }$ and $M_2=135_{-18}^{+19}{M}_{\oplus }$. Both planets have Saturn-like densities (${\rho }_1=0.73_{-0.13}^{+0.14}{\text{g cm}}^{-3}$, ${\rho }_2=0.66_{-0.23}^{+0.41}{\text{g cm}}^{-3}$), which appears to be a growing trend among GEMS systems and, more generally, warm Jupiters. In confirming both of these exoplanets, we add to the growing evidence for a population of Saturn-density planets among the GEMS systems. We also find evidence for a preliminary trend in which GEMS exhibit systematically closer orbits compared to FGK giants.