On the value distribution of positive definite quadratic forms

摘要

Denote by 0 = λ0 < λ1 ≤ λ2 ≤ . . . the infinite sequence given by the values of a positive definite irrational quadratic form in k variables at integer points. For l ≥ 2 and an (l -1)-dimensional interval I = I2×. . .×Il we consider the l-level correlation function K(l)I (R) which counts the number of tuples (i1, . . ., il) such that λi1,...,λil ≤ R2 and λij - λi1 ∈ Ij for 2 ≤ j ≤ l. We study the asymptotic behavior of K(l)I (R) as R tends to infinity. If k ≥ 4 we prove K(l)I (R) ~ cl (Q) vol (I) Rlk-2(l-1) for arbitrary l, where cl (Q) is an explicitly determined constant. This remains true for k = 3 under the restriction l ≤ 3.