The fractal nature of the functional law of logarithm of fractional Brownian motions (Q1764999)

Let \(\{ X( t) ,t\geq 0\} \) be a fractional Brownian motion with \(0<\alpha <1\) and \[ M_{t,h}( x) =\frac{X( t+hx) -X( t) }{h^{\alpha }( 2\max ( 1,\ln x) ) ^{1/2}},\quad 0<x<1,\quad 0<t<1-h,\quad 0<h<1. \] Let \(\mu = ( X) \) and \(U=\{ f\in H_{\mu }:\| f\| _{\mu }\leq 1\} \) where \(H_{\mu }\subset C_{0}[ 0,1 ] \) is the reproducing Hilbert space. The main statement is as follows. Given an analytic set \(E\subset [ 0,1] \) and a function \( f\in U,\) if \(\| f\| ^{2}\) is less than the packing dimension \(\dim _{p}( E) , \) then a.s. there exists a time \(t\in E \) such that the normalized increment process \(M_{t,h}( x) \) converges uniformly to \(f\) along a sequence of \(h\) values tending to zero, but if \(\| f\| _{\mu }^{2}>\dim _{p}( E) \), then a.s. there is no such time in \(E\).