On the value distribution of positive definite quadratic forms

摘要

Denote by 0 = λ ₀ < λ ₁ ≤ λ ₂ ≤ . . . 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 = I ₂×. . .×I _l we consider the l-level correlation function K(l)_I(R){K^{(l)}_I(R)} which counts the number of tuples (i ₁, . . . , i _l ) such that l_i1,¼,l_il £ R2{lambda_{i_1},ldots,lambda_{i_l}leq R^2} and l_ij-l_i1 Î I_j{lambda_{i_{j}}-lambda_{i_{1}}in I_j} for 2 ≤ j ≤ l. We study the asymptotic behavior of K(l)_I(R){K^{(l)}_I(R)} as R tends to infinity. If k ≥ 4 we prove K(l)_I(R) ~ c_l(Q) vol(I)Rlk-2(l-1){K^{(l)}_I(R)sim c_l(Q),{rm vol}(I)R^{lk-2(l-1)}} for arbitrary l, where c _l (Q) is an explicitly determined constant. This remains true for k = 3 under the restriction l ≤ 3.