English: Probability distributions related to Thomae's function can be derived from recurrent processes generated from uniform discrete distributions. Such uniform discrete distributions can be pi digits, flips of a fair dice or live casino spins. In greater detail, the recurrent process is characterized as follows: A random variable C_i is repeatedly sampled N times from a discrete uniform distribution, where i ranges from 1 to N. For instance, consider integer values ranging from 1 to 10. Moments of occurrence, T_k, signify when events Ci repeat, defined as C_i = C_i-1 or C_i = C_i-2, where k ranges from 1 to M, with M being less than N. Subsequently, define S_j as the interval between successive T_k, representing the waiting time for an event to occur. Finally, introduce Zl as ln(S_j) – ln(S_j-1), where l ranges from 1 to U-1. The random variable Z displays fractal properties, resembling the shape distribution akin to Thomae's or Dirichlet function.