  We exhibit a family of autosimilar H\"older maps that satisfies a fractal
version of the Van Der Corput Lemma, despite not being absolutely continuous.
The result is a direct consequence of a recent work of Sahlsten and Steven
arXiv:2009.01703, which is based on a powerful theorem of Bourgain known as a
sum-product phenomenon estimate. We give a substantially simpler proof of this
fact in our particular context, using an elementary method inspired from
arXiv:1704.02909 to check the non-concentration estimates that are needed to
apply the sum-product phenomenon. This method allows us to gain additional
control over the decay rate.
