Choices, intervals and equidistribution

TL;DR

This paper establishes a sufficient condition for the equidistribution of sequences generated by $ ext{Psi}$-processes in [0,1], including the $ ext{max}$-2 process, solving an open problem and extending to more general processes.

Contribution

It provides a new sufficient condition for equidistribution of $ ext{Psi}$-process sequences, including the $ ext{max}$-2 process, and extends results to biased interpolations.

Findings

The $ ext{max}$-2 process produces equidistributed sequences.

A sufficient condition for equidistribution in $ ext{Psi}$-processes is identified.

Equidistribution is established for a class of interpolated $ ext{Psi}$-processes.

Abstract

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 $n$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 $\Psi$-processes. This includes an interpolation of the $\min$-2 and $\max$-2 processes that is biased towards $\min$-2.