On pair correlations and Hausdorff dimension |
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Authors: | R. Nair |
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Affiliation: | (1) Mathematical Sciences, The Univerisity of Liverpool, Liverpool, L697ZL, U.K. |
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Abstract: | We consider a system of “generalised linear forms” defined at a point x = (x (i, j)) in a subset of R d by for k ≥ 1. Here d = d 1 + ⋯ + d l and for each pair of integers (i, j) ∈ D, where D = {(i, j): 1 ≤ i ≤ l, 1 ≤ j ≤ d i } the sequence of functions (g (i, j), k (x)) k=1∞ are differentiable on an interval X ij contained in R. We study the distribution of the sequence on the l-torus defined by the fractional parts X k (x) = ({ L 1(x)(k)}, ..., {L l (x)(k)}) ∈ T l , for typical x in the Cartesian product . More precisely, let R = I 1 × ⋯ × I l be a rectangle in T l and for each N ≥ 1 define a pair correlation function and a discrepancy , where the supremum is over all rectangles in T l and χ R is the characteristic function of the set R. We give conditions on (g (i, j), k (x)) k=1∞ to ensure that given ε > 0, for almost every x ∈ T l we have Δ N (x) = o(N(log N) l+∈). Under related conditions on(g (i, j), k (x)) k =1∞ we calculate for appropriate β ∈ (0, 1) the Hausdorff dimension of the set {x : lim sup N→∞ N β Δ N (x > 0)}. Our results complement those of Rudnick and Sarnak and Berkes, Philipp, and Tichy in one dimension and M. Pollicott and the author in higher dimensions. |
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