  We analyse a random walk on the ring of integers mod $n$, which at each time
point can make an additive `step' or a multiplicative `jump'. When the
probability of making a jump tends to zero as an appropriate power of $n$ we
prove the existence of a total variation pre-cutoff for this walk. In addition,
we show that the process obtained by subsampling our walk at jump times
exhibits a true cutoff, with mixing time dependent on whether the step
distribution has zero mean.
