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Localized quantitative criteria for equidistribution. (English) Zbl 1381.11059

An infinite sequence \((x_n)_{n\geq 1}\) in \([0,1)\) is said to have Poissonian pair correlations if \[ \lim_{N\rightarrow\infty}\frac{1}{N} \# \left\{1\leq l\neq m\leq N:\, | x_l - x_m | <\frac{s}{N}\right\}=2s \] for all positive \(s\). It is known from results by C. Aistleitner et al. [J. Number Theory 182, 206–220 (2018; Zbl 1415.11106)], as well as S. Grepstad and G. Larcher [Arch. Math. 109, No. 2, 143–149 (2017; Zbl 1387.11064)] that sequences with Poissonian pair correlation are uniformly distributed.
The author of this paper generalizes this result and shows that a whole family of related criteria guarantee uniform distribution of a sequence. This generalization is shown on arbitrary compact manifolds and recovers the aforementioned result. The main result in the paper is as follows.
Let \((M,g)\) be a smooth compact manifold and let \((x_n)_{n\geq 1}\) be a sequence on \(M\). If there exists a bounded sequence of \(0< t_N \leq C\) such that \[ \lim_{N\rightarrow \infty}\frac{1}{N^2} \sum_{m,n=1}^N \left(e^{t_N \Delta}\delta_{x_m}\right) (x_n)=\frac{1}{\mathrm{vol} (M)}, \] then \((x_n)_{n\geq 1}\) is uniformly distributed on \((M,g)\). Here \(e^{t\Delta}\) denotes the heat kernel. If \(t_N=c\) is constant, this condition is even equivalent to uniform distribution.

MSC:

11K06 General theory of distribution modulo \(1\)
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
42A82 Positive definite functions in one variable harmonic analysis
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