Choices, intervals and equidistribution

We give a sufficient condition for a random sequence in [0,1] generated by a Psi-process to be equidistributed. The condition is met by the canonical example - the max-2 process - where the nth 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 Psi-processes. This includes an interpolation of the min-2 and max-2 processes that is biased towards min-2.