We determine the Hausdorff dimension of a particle path, $D_{\rm H}$, in the
    recently proposed `smeared space’ model of quantum geometry. The model
    introduces additional degrees of freedom to describe the quantum state of the
    background and gives rise to both the generalised uncertainty principle (GUP)
    and extended uncertainty principle (EUP) without introducing modified
    commutation relations. We compare our results to previous studies of the
    Hausdorff dimension in GUP models based on modified commutators and show that
    the minimum length enters the relevant formulae in a different way. We then
    determine the Hausdorff dimension of the particle path in smeared momentum
    space, $\tilde{D}_{\rm H}$, and show that the minimum momentum is dual to the
    minimum length. For sufficiently coarse grained paths, $D_{\rm H} =
    \tilde{D}_{\rm H} = 2$, as in canonical quantum mechanics. However, as the
    resolutions approach the minimum scales, the dimensions of the paths in each
    representation differ, in contrast to their counterparts in the canonical
    theory. The GUP-induced corrections increase $D_{\rm H}$ whereas the
    EUP-induced corrections decrease $\tilde{D}_{\rm H}$, relative to their
    canonical values, and the extremal case corresponds to $D_{\rm H} = 3$,
    $\tilde{D}_{\rm H} = 1$. These results show that the GUP and the EUP affect the
    fractal properties of the particle path in fundamentally different, yet
    complimentary, ways.



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