Intermediate dimension of images of sequences under fractional Brownian motion

Abstract

We show that the almost sure $\theta$-intermediate dimension of the image of the set $F_p =\{0, 1,\frac{1}{2^p},\frac{1}{3^p},\ldots\}$ under index-$h$ fractional Brownian motion is $\frac{\theta}{ph+\theta}$, a value that is smaller than that given by directly applying the H\"{o}lder bound for fractional Brownian motion. In particular this establishes the box-counting dimension of these images.