Localized quantitative criteria for equidistribution
Volume 180 / 2017
Abstract
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 $(t_n)_{n=1}^{\infty}$ of positive real numbers converging to 0 such that $$ \lim_{N \rightarrow \infty}{ \frac{1}{N^2} \sum_{m,n = 1}^{N}{ \frac{1}{\sqrt{t_N}} \exp\biggl(- \frac{1}{t_N} (x_m - x_n)^2 \biggr)} } = \sqrt{\pi}, $$ then $(x_n)_{n=1}^{\infty}$ is uniformly distributed. This is especially interesting when $t_N \sim N^{-2}$ since the size of the sum is then essentially determined by local gaps at scale $\sim N^{-1}$. This can 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 $e^{t\varDelta}$: for all $x_1, \dots, x_N \in M$ and all $t \gt 0$ we have $$ \frac{1}{N^2}\sum_{m,n=1}^{N}{[e^{t\varDelta}\delta_{x_m}](x_n)} \geq \frac{1}{\mvol(M)}, $$ and equality is attained as $N \rightarrow \infty$ if and only if $(x_n)_{n=1}^{\infty}$ equidistributes.
