The uniform distribution is complete with respect to testing identity to a fixed distribution

Inspired by Diakonikolas and Kane (2016), we reduce the class of problems consisting of testing whether an unknown distribution over [n] equals a fixed distribution to this very problem when the fixed distribution is uniform over [n]. Our reduction preserves the parameters of the problem, which are n and the proximity parameter ǫ > 0, up to a constant factor. While this reduction yields no new bounds on the sample complexity of either problems, it provides a simple way of obtaining testers for equality to arbitrary fixed distributions from testers for the uniform distribution.