Simultaneous Diophantine approximation of rationals by rationals

Let α = (a1B,…, anB) be a vector of rational numbers satisfying the primitivity condition g.c.d. (a1,…, an, B) = 1. This paper studies the number N(α, Δ) of simultaneous Diophantine approximations to α with denominators x < B of a given degree of approximation measured by Δ, i.e., N(α, Δ) is the number of vectors ξ = (x1x,…, xnx) with 1 ≦ x < B such that |aiB − xix| ≦ ΔBx for 1 ≦ i ≦ n. It gives estimates for the first and second moments of N(α, Δ) over the ensemble Sn(B) consisting of all primitive vectors α in the unit n-cube having denominator B. As a consequence it shows for n ≧ 5 that “most” vectors in Sn(B) that have one “unusually good” simultaneous Diophantine approximation have a bounded number of such approximations. The paper also estimates the moments of the number of solutions f(λ, B, Δ1, Δ2) to the homogenous linear congruence λx1 ≡ x2 (mod B) with bounds |x1| ≦ Δ1, |x2| ≦ Δ2 on the variables, taken over the set of λ with (λ, B) = 1.