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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. 219-238. doi:10.1007/s11856-006-0009-z ISSN 0021-2172.
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Official URL: http://dx.doi.org/10.1007/s11856-006-0009-z
Abstract
We consider a system of "generalised linear forms" defined on a subset x = (x(ij)) of R-d by
L-1 (x) (k) = d(1) g(1j)(k) (x(1j)), ...L-l (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 X-ij. Then let
X-K(x) = ({L-1(x)(k)}, . . . , {L-l(x)(k)}) is an element of T-l
for x in the Cartesin product X = x(i=1)(l) x (di)(j = 1) X-ij subset of R-d. Let R = I-1 x . . . x I-l be a rectangle in T-l and for each N >= 1 let
VN(R) = Sigma(1 <= n not equal m <= N) XR(X-n)(X)-(X-m)(x))
and then define
Delta(N) = sup {V-N (R) - N (N-1) leb (R) } R subset of T-l
where the supremum is over all rectangles in T-l. We show that for almost every x is an element of T-d we have that
Delta(N) = O(N(log N)(alpha))
for appropiate alpha. Other related results are also described.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Israel Journal of Mathematics | ||||
Publisher: | Magnes Press | ||||
ISSN: | 0021-2172 | ||||
Official Date: | January 2007 | ||||
Dates: |
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Volume: | Volume 157 | ||||
Number: | Number 1 | ||||
Number of Pages: | 20 | ||||
Page Range: | pp. 219-238 | ||||
DOI: | 10.1007/s11856-006-0009-z | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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