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Pair correlations of sequences in higher dimensions
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Nair, R and Pollicott, Mark (2007) Pair correlations of sequences in higher dimensions. Israel Journal of Mathematics, Volume 157 (Number 1). pp. 219238. doi:10.1007/s118560060009z
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Official URL: http://dx.doi.org/10.1007/s118560060009z
Abstract
We consider a system of "generalised linear forms" defined on a subset x = (x(ij)) of Rd by
L1 (x) (k) = d(1) g(1j)(k) (x(1j)), ...Ll (x) (k) = d(l) g(lj)(k) (xlj) is an element of R, for k >= 1,
where d = d(1) + ...+ d(l) and for each pair of integers (i, j), 1 <= i <= l l <= j <= d(i) the sequence of functions (g(ij)(k)(x))(kappa=1)(infinity) is differentiable on an interval Xij. Then let
XK(x) = ({L1(x)(k)}, . . . , {Ll(x)(k)}) is an element of Tl
for x in the Cartesin product X = x(i=1)(l) x (di)(j = 1) Xij subset of Rd. Let R = I1 x . . . x Il be a rectangle in Tl and for each N >= 1 let
VN(R) = Sigma(1 <= n not equal m <= N) XR(Xn)(X)(Xm)(x))
and then define
Delta(N) = sup {VN (R)  N (N1) leb (R) } R subset of Tl
where the supremum is over all rectangles in Tl. We show that for almost every x is an element of Td we have that
Delta(N) = O(N(log N)(alpha))
for appropiate alpha. Other related results are also described.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics  
Divisions:  Faculty of Science > Mathematics  
Journal or Publication Title:  Israel Journal of Mathematics  
Publisher:  Magnes Press  
ISSN:  00212172  
Official Date:  January 2007  
Dates: 


Volume:  Volume 157  
Number:  Number 1  
Number of Pages:  20  
Page Range:  pp. 219238  
DOI:  10.1007/s118560060009z  
Status:  Peer Reviewed  
Publication Status:  Published  
Access rights to Published version:  Restricted or Subscription Access 
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