Thin ultrafilters and the P-hierarchy of ultrafilters

Under Martin Axiom, we prove that for each ordinal γ<ω₁ there exists a thin ultrafilter that belongs to the class P_γ of the P-hierarchy of ultrafilters. Since the class P₂ of ultrafilters coincides with the class of P-points, this result generalizes a theorem of Flašková, which states that, under the Martin Axiom for countable posets, there exists a thin ultrafilter which is not a P-point. It is also related to a theorem which states that, under Continuum Hypothesis, for any tall P-ideal I on ω there are I-ultrafilters in each class P_γ of the P-hierarchy. However, the ideal of thin sets is not a P-ideal.