  The site-diluted compound (Yb$_{1-x}$Lu$_x$)$_4$As$_3$ is a scarce
realization of the linear Heisenberg antiferromagnet partitioned into
finite-size segments and is an ideal model compound for studying
field-dependent effects of quenched disorder in the one-dimensional
antiferromagnets. It differentiates from the systems studied so far in two
aspects - the type of randomness and the nature of the energy gap in the pure
sample. We have measured the specific heat of single-crystal
(Yb$_{1-x}$Lu$_x$)$_4$As$_3$ in magnetic fields up to 19.5 T. The contribution
$C_{\perp}$ arising from the magnetic subsystem in an applied magnetic field
perpendicular to the chains is determined. Compared to pure Yb$_4$As$_3$, for
which $C_{\perp}$ indicates a gap opening, for diluted systems a
non-exponential decay is found at low temperatures which is consistent with the
thermodynamic scaling of the specific heat established for a Bose-glass phase.
