If very frequent periodic measurements ascertain whether a quantum system is still in its initial state, its evolution is hindered. This peculiar phenomenon is called quantum Zeno effect. We investigate the large-time limit of the survival probability as the total observation time scales as a power of the measurement frequency, $t∝ N^α$. The limit survival probability exhibits a sudden jump from 1 to 0 at $α=1/2$, the threshold between the quantum Zeno effect and a diffusive behavior. Moreover, we show that for $α≥ 1$ the limit probability becomes sensitive to the spectral properties of the initial state and to arithmetic properties of the measurement periods. ; Comment: 18 pages, 1 figure