Israel Journal of Mathematics

, Volume 157, Issue 1, pp 219–238

# Pair correlations of sequences in higher dimensions

• R. Nair
• M. Pollicott
Article

## Abstract

We consider a system of “generalised linear forms” defined on a subset x = (x ij ) of ℝd by
$$L_1 (\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{x} )(k) = \sum\limits_{j = 1}^{d_1 } {g_{1j}^k (x_{1j} ), \ldots ,} L_l (\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{x} )(k) = \sum\limits_{j = 1}^{d_l } {g_{lj}^k (x_{lj} ) \in \mathbb{R}} , for k \geqslant 1,$$
, where d = d 1 + ⋯+ d l and for each pair of integers (i, j), 1 ≤ il, 1 ≤ jd i the sequence of functions (g ij k (x)) k=1 is differentiable on an interval X ij . Then let
$$X_k (\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{x} ) = (\{ L_1 (\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{x} )(k)\} , \ldots ,\{ L_l (\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{x} )(k)\} ) \in \mathbb{T}^l ,$$
, for x in the Cartesian product X = × i=1 l × j=1 di X ij ⊂ ℝd. Let R = I 1 × ⋯ × I l be a rectangle in $$\mathbb{T}^l$$ and for each N ≥ 1 let
$$V_N (R) = \sum\limits_{1 \leqslant n \ne m \leqslant N} {\chi _R (\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{X} _n (\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{x} ) - \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{X} _m (\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{x} ))}$$
and then define
$$\Delta _N = \mathop {\sup }\limits_{R \subset \mathbb{T}^l } \{ V_N (R) - N(N - 1) leb(R)\}$$
where the supremum is over all rectangles in $$\mathbb{T}^l$$. We show that for almost every x$$\mathbb{T}^d$$ we have that
$$\Delta _N = O(N(\log N)^\alpha ),$$
for appropriate α. Other related results are also described.

### Keywords

High Dimension Pair Correlation London Mathematical Society Dependent Random Variable Half Open Interval

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