On pair correlations and Hausdorff dimension

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.