Localized Quantitative Criteria for Equidistribution

Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). We show that if there exists a sequence of positive real numbers $(t_n)_{n=1}^{\infty}$ converging to 0 such that $$\lim_{N \rightarrow \infty}{ \frac{1}{N^2} \sum_{m,n = 1}^{N}{ \frac{1}{\sqrt{t_N}} \exp{\left(- \frac{1}{t_N} (x_m - x_n)^2 \right)}} } = \sqrt{\pi},$$ then $(x_n)_{n=1}^{\infty}$ is uniformly distributed. This is especially interesting when $t_N$ is close to $N^{-2}$ since the size of the sum is then mostly determined by local gaps at scale $\sim N^{-1}$. A similar argument can then be used to show equidistribution of sequences with Poissonian pair correlation, which recovers a recent result of Aistleitner, Lachmann \& Pausinger and Grepstad \& Larcher. The general form of the result is proven on arbitrary compact manifolds $(M,g)$ where the role of the exponential function is played by the heat kernel $e^{t\Delta}$: for all $x_1, \dots, x_N \in M$ and all $t>0$ $$\frac{1}{N^2} \sum_{m,n=1}^N {[e^{t\Delta}\delta_{x_m}](x_n)} \geq \frac{1}{vol(M)}$$ and equality is attained as $N \rightarrow \infty$ if and only if $(x_n)_{n=1}^\infty$ equidistributes.