Title: Localized Quantitative Criteria for Equidistribution,
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 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 $t_N \sim N^{-2}$ since the size of the sum is then
essentially determined exclusively 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 $e^{t\Delta}$: 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.