Hilbert-Kunz series, F-signature series, and weak p-fractals

We extend the theory of $p$-fractals of Monsky and Teixeira by introducing the notion of weak $p$-fractal. We prove that for a hypersurface $f$ having rational Hilbert-Kunz series is equivalent to the weak $p$-fractality of the associated function ${\phi }_{f,p}$ and having rational F-signature series is equivalent to the weak $p$-fractality of the reflection ${\bar{\phi }}_{f,p}$. In addition, we prove some results characterizing the shape of the generating series of numerical functions which are quasi-polynomials in $p^n$. This is motivated by the fact that the Hilbert-Kunz and F-signature functions take this form in several examples of interest.