A Lower Bound on the Complexity of Testing Grained Distributions

For a natural number m , a distribution is called m -grained, if each element appears with probability that is an integer multiple of 1/m . We prove that, for any constant c<1 , testing whether a distribution over [Θ(m)] is m -grained requires Ω(m^c) samples, where testing a property of distributions means distinguishing between distributions that have the property and distributions that are far (in total variation distance) from any distribution that has the property.