Choices, intervals and equidistribution

We give a sufficient condition for a random sequence in [0,1] generated by a $\mathrm{\Psi }$-process to be equidistributed. The condition is met by the canonical example – the $\max$-2 process – where the $n$th term is whichever of two uniformly placed points falls in the larger gap formed by the previous $n-1$ points. This solves an open problem from Itai Benjamini, Pascal Maillard and Elliot Paquette. We also deduce equidistribution for more general $\mathrm{\Psi }$-processes. This includes an interpolation of the $\min$-2 and $\max$-2 processes that is biased towards $\min$-2.