Pair correlations of the leVeque sequence on the polydisc

Abstract

AbstractWe consider a system of “forms” defined for ẕ = (zij) on a subset of $\Bbb C^d$ by where d = d1 + ⋅ ⋅ ⋅ + dl and for each pair of integers (i,j) with 1 ≤ i ≤ l, 1 ≤ j ≤ di we denote by $(v_{ij}(k))_{k=1}^{\infty}$ a strictly increasing sequence of natural numbers. Let ${\Bbb C}_1$ = {z ∈ ${\Bbb C}$ : |z| < 1} and let ${\underline X} \ = \ \times _{i=1}^l \times _{j=1}^{d_i}X_{ij}$ where for each pair (i, j) we have Xij = ${\Bbb C}\backslash {\Bbb C}_1$. We study the distribution of the sequence on the l-polydisc $({\Bbb C}_1)^l$ defined by the coordinatewise polar fractional parts of the sequence Xk(ẕ) = (L1(ẕ)(k),. . ., Ll(ẕ)(k)) for typical ẕ in ${\underline X}$ More precisely for arcs I1, . . ., I2l in $\Bbb T$, let B = I1 × ⋅ ⋅ ⋅ × I2l be a box in $\Bbb T^{2l}$ and for each N ≥ 1 define a pair correlation function by and a discrepancy by ΔN = $\sup_{B \subset \Bbb T^{2l}}${VN(B) − N(N−1)leb(B)}, where the supremum is over all boxes in $\Bbb T^{2l}$. We show, subject to a non-resonance condition on $(v_{ij}(k))_{k=1}^{\infty}$, that given ε > 0 we have ΔN = o(N$(log N)^{l + {1\over 2}}$(log log N)1+ε) for almost every $\underline x(\underline z)\in \Bbb T^{2l}$. Similar results on extremal discrepancy are also proved. Our results complement those of I. Berkes, W. Philipp, M. Pollicott, Z. Rudnick, P. Sarnak, R Tichy and the author in the real setting.