Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Poissonian Pair Correlation in Higher Dimensions (1812.10458v3)

Published 26 Dec 2018 in math.CA and math.NT

Abstract: Let $(x_n){n=1}{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). A sequence $(x_n)$ is said to have Poissonian pair correlation if, for all $s>0$, $$ \lim{N \rightarrow \infty}{ \frac{1}{N} # \left{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right}} = 2s.$$ It is known that this implies uniform distribution of the sequence $(x_n)$. Hinrichs, Kaltenb\"ock, Larcher, Stockinger & Ullrich extended this result to higher dimensions and showed that sequences $(x_n)$ in $[0,1]d$ that satisfy, for all $s>0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} # \left{ 1 \leq m \neq n \leq N: |x_m - x_n|{\infty} \leq \frac{s}{N} \right}} = (2s)d.$$ are also uniformly distributed. We prove the same result for the extension by the Euclidean norm: if a sequence $(x_n)$ in $\mathbb{T}d$ satisfies, for all $s > 0$, $$ \lim{N \rightarrow \infty}{ \frac{1}{N} # \left{ 1 \leq m \neq n \leq N: |x_m - x_n|_{2} \leq \frac{s}{N} \right}} = \omega_d sd$$ where $\omega_d$ is the volume of the unit ball, then $(x_n)$ is uniformly distributed. Our approach shows that Poissonian Pair Correlation implies an exponential sum estimate that resembles and implies the Weyl criterion.

Summary

We haven't generated a summary for this paper yet.