Localized Quantitative Criteria for Equidistribution

Let $(x_n{)}_{n=1}^{\mathrm{\infty }}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). We show that if there exists a sequence of positive real numbers $(t_n{)}_{n=1}^{\mathrm{\infty }}$ converging to 0 such that $$\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N^2}\sum _{m,n=1}^N\frac{1}{\sqrt{t_N}}\exp (-\frac{1}{t_N}(x_m-x_n{)}^2)=\sqrt{\pi },$$ then $(x_n{)}_{n=1}^{\mathrm{\infty }}$ 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 $e^{t\mathrm{\Delta }}$: for all $x_1,\dots ,x_N\in M$ and all $t>0$ $$\frac{1}{N^2}\sum _{m,n=1}^N[e^{t\mathrm{\Delta }}{\delta }_{x_m}](x_n)\ge \frac{1}{vol(M)}$$ and equality is attained as $N\to \mathrm{\infty }$ if and only if $(x_n{)}_{n=1}^{\mathrm{\infty }}$ equidistributes.