Uniformity Testing over Hypergrids with Subcube Conditioning

We give an algorithm for testing uniformity of distributions supported on hypergrids $[m]^n$, which makes $\tilde{O}(\text{poly}(m)\sqrt{n}/\epsilon^2)$ queries to a subcube conditional sampling oracle. When the side length $m$ of the hypergrid is a constant, our algorithm is nearly optimal and strengthens the algorithm of [CCK+21] which has the same query complexity but works for hypercubes $\{\pm 1\}^n$ only. A key technical contribution behind the analysis of our algorithm is a proof of a robust version of Pisier's inequality for functions over $\mathbb{Z}_m^n$ using Fourier analysis.