  We regard the classic Thue--Morse diffraction measure as an equilibrium
measure for a potential function with a logarithmic singularity over the
doubling map. Our focus is on unusually fast scaling of the Birkhoff sums
(superlinear) and of the local measure decay (superpolynomial). For several
scaling functions, we show that points with this behavior are abundant in the
sense of full Hausdorff dimension. At the fastest possible scaling, the
corresponding rates reveal several remarkable phenomena. There is a gap between
level sets for dyadic rationals and non-dyadic points, and beyond dyadic
rationals, non-zero accumulation points occur only within intervals of positive
length. The dependence between the smallest and the largest accumulation point
also manifests itself in a non-trivial joint dimension spectrum.
