  Recent low temperature heat capacity (C$_P$) measurements on polycrystalline
samples of the pyrochlore antiferromagnet Tb$_{2+x}$Ti$_{2-x}$O$_{7+\delta}$
have shown a strong sensitivity to the precise Tb concentration $x$, with a
large anomaly exhibited for $x \sim 0.005$ at $T_C \sim 0.5$ K and no such
anomaly and corresponding phase transition for $x \le 0$. We have grown single
crystal samples of Tb$_{2+x}$Ti$_{2-x}$O$_{7+\delta}$, with approximate
composition $x=-0.001, +0.0042$, and $+0.0147$, where the $x=0.0042$ single
crystal exhibits a large C$_P$ anomaly at $T_C$=0.45 K, but neither the
$x=-0.001$ nor the $x=+0.0147$ single crystals display any such anomaly. We
present new time-of-flight neutron scattering measurements on the $x=-0.001$
and the $x=+0.0147$ samples which show strong
$\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$ quasi-Bragg peaks at low
temperatures characteristic of short range antiferromagnetic spin ice (AFSI)
order at zero magnetic field but only under field-cooled conditions, as was
previously observed in our $x = 0.0042$ single crystal. These results show that
the strong $\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$ quasi-Bragg peaks
and gapped AFSI state at low temperatures under field cooled conditions are
robust features of Tb$_2$Ti$_2$O$_7$, and are not correlated with the presence
or absence of the C$_P$ anomaly and phase transition at low temperatures.
Further, these results show that the ordered state giving rise to the C$_P$
anomaly is confined to $0 \leq x \leq 0.01$ for
Tb$_{2+x}$Ti$_{2-x}$O$_{7+\delta}$, and is not obviously connected with
conventional order of magnetic dipole degrees of freedom.
