ON OSCILLATORY INTEGRALS WITH HOLDER PHASES

We exhibit a family of autosimilar Holder maps that satisfies a "fractal" version of the Van Der Corput Lemma, despite not being absolutely continuous. Those maps are related to the measure of maximal entropy for some expanding dynamics on the circle. This result is a direct consequence of a recent work of Sahlsten and Steven (Amer. J. Math), 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 Bourgain and Dyatlov's earlier work (GAFA) 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.