  Due to the nonlinear dielectric response within SrTiO$_{3}$ (STO), an
accumulation layer created by positive charges at the surface of the STO sample
($x=0$) has an electron density profile $n(x)$ that slowly decays as
$1/x^{12/7}$. Here we show that the long tail of $n(x)$ causes the
magnetization and the specific heat of the accumulation layer to diverge at
large $x$. We explore the truncation of the tail by the finite sample width
$W$, the transition from the nonlinear to linear dielectric response with
dielectric constant $\kappa$, and the use of a back gate with a negative
voltage $-|V|$. We find that both the magnetization and specific heat are
anomalously large and obey nontrivial power law dependences on $W$, $\kappa$,
or $|V|$. We conclude with a discussion of how the capacitance as a function of
the back gate voltage may be used to study the shape of the $n(x)$ tail in thin
samples.
