Local fluctuations of critical Mandelbrot cascades

We investigate so-called generalized Mandelbrot cascades at the freezing (critical) temperature. It is known that, after a proper rescaling, a~sequence of multiplicative cascades converges weakly to some continuous random measure. Our main question is how the limiting measure $\mu$ fluctuates. For any given point $x$, denoting by $B_n(x)$ the ball of radius $2^{-n}$ centered around $x$, we present optimal lower and upper estimates of $\mu (B_n(x))$ as $n \to \infty$.