Optimal Closeness Testing of Discrete Distributions Made (Complex) Simple

In this note, we revisit the recent work of Diakonikolas, Gouleakis, Kane, Peebles, and Price (2021), and provide an alternative proof of their main result. Our argument does not rely on any specific property of Poisson random variables (such as stability and divisibility) nor on any "clever trick," but instead on an identity relating the expectation of the absolute value of any random variable to the integral of its characteristic function: $$\mathbb{E}[|X|] = \frac{2}{\pi}\int_0^\infty \frac{1-\Re(\mathbb{E}[e^{i tX}])}{t^2}\, dt$$ Our argument, while not devoid of technical aspects, is arguably conceptually simpler and more general; and we hope this technique can find additional applications in distribution testing.