  For i.i.d. random vectors $(M_{1},Q_{1}),(M_{2},Q_{2}),\ldots$ such that
$M>0$ a.s., $Q\geq 0$ a.s. and $\mathbb{P}(Q=0)<1$, the random difference
equation $X_{n}=M_{n}X_{n-1}+Q_{n}$, $n=1,2,\ldots$, is studied in the critical
case when the random walk with increments $\log M_{1},\log M_{2}$ is
oscillating. We provide conditions for the null-recurrence and transience of
the Markov chain $(X_{n})_{n\ge 0}$ by inter alia drawing on techniques
developed in the related article Alsmeyer et al (2017) for another case
exhibiting the null-recurrence/transience dichotomy.
