Localized quantitative criteria for equidistribution

Stefan Steinerberger¹

¹ Department of Mathematics Yale University New Haven, CT 06511, U.S.A.

Acta Arithmetica, Tome 180 (2017) no. 2, pp. 183-199

Résumé

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.

Détail

DOI : 10.4064/aa170410-22-5

Keywords: infty sequence torus mathbb normalized length nbsp there exists sequence infty positive real numbers converging lim rightarrow infty frac sum frac sqrt exp biggl frac biggr sqrt infty uniformly distributed especially interesting sim since size sum essentially determined local gaps scale sim equidistribution sequences poissonian pair correlation which recovers recent result aistleitner lachmann pausinger grepstad larcher general form result proven arbitrary compact manifolds where role exponential function played heat kernel vardelta dots have frac sum vardelta delta geq frac mvol equality attained rightarrow infty only infty equidistributes