On pair correlations and Hausdorff dimension

Abstract

We consider a system of “generalised linear forms” defined at a point x = (x _{(i, j)}) in a subset of R ^d by

$$ L_i (x)(k) = \sum\limits_{j = 1}^{d_i } {g_{(i,j),k} (x_{(i,j)} )} \in R, i = 1,2, \ldots $$

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 $X = \prod\nolimits_{i = 1}^l {\prod\nolimits_{j = 1}^{d_i } {X_{ij} \subseteq R^d } }$. More precisely, let R = I ₁ × ⋯ × I _l be a rectangle in T ^l and for each N ≥ 1 define a pair correlation function

$$ V_N (R)(x) = \sum\limits_{1 \leqslant n \ne m \leqslant N} {\chi _R } (X_n (x) - X_m (x)) $$

and a discrepancy $\Delta _N (x) = \sup _{R \subseteq {\rm T}^l } \{ V_N (R)(x) - N(N - 1) leb(R)\}$, 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.