Enhanced Fluctuation/Inhomogeneity Effects Near The Superconductor-Insulator Transition in Severely Underdoped 2D Films of Ca$_{0.3}$Y$_{0.7}$Ba$_2$Cu$_3$O$_{7-\delta}$

It has been shown that the superconductor-insulator transition in two-dimensional, severely underdoped films of Ca-doped YBCO is accompanied by a linear scaling between the zero temperature superfluid density $n_s(T=0) $ and the temperature for the unbinding of thermally excited vortex pairs, $T_{2D}$.\footnote{I. Hetel, T.R. Lemberger and M. Randeria, Nature Phys {\textbf 3}, 700 (2007).} We present results from resistivity and magnetic screening studies on 2D Ca-YBCO showing a linear relationship between the width of tails in the resistive transition and $T_{2D}$. These tails show a significant suppression of resistivity and can extend down for tens of Kelvins before the films exhibit magnetic screening. We examine this relationship in light of a possible enhancement of inhomogeneous effects in the vicinity of the superconductor-insulator transition with underdoping.