I will introduce two families of statistical tests aiming at testing uniformity of samples of data points on homogeneous compact Polish spaces. Such tests are based on the computation of nearest neighbours distances, as in [1]. Such tests are consistent and come with parametric separation rates. I will show numerical results on the flat torus, the circle, the sphere, and a subset of the Poincaré disk. In particular, the tests will be compared to classical tests on the sphere and the circle [2].

[1] Brécheteau, A statistical test of isomorphism between metric-measure spaces using the distance-to-a-measure signature, 2019

[2] Garciá-Portugués Verdebout, An overview of uniformity tests on the hypersphere, 2018