Limits of fractality: Zeno boxes and relativistic particles

Physical fractals invariably have upper and lower limits for their fractal structure. Berry has shown that a particle sharply confined to a box has a wave function that is fractal both in time and space, with no lower limit. In this article, two idealizations of this picture are softened and a corresponding lower bound for fractality obtained. For a box created by repeated measurements (\`a la the quantum Zeno effect), the lower bound is $\Delta x\sim \Delta t (\hbar/{mL})$ with $\Dt$ the interval between measurements and $L$ is the size of the box. For a relativistic particle, the lower bound is the Compton wavelength, $\hbar/mc$. The key step in deriving both results is to write the propagator as a sum over classical paths.