Simultaneous Diophantine approximation of rationals by rationals

Abstract

Let α = ( a 1 B ,…, a n B ) be a vector of rational numbers satisfying the primitivity condition g.c.d. ( a 1,…, a n , 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 = ( x 1 x ,…, x n x ) with 1 ≦ x < B such that | a i B − x i x | ≦ Δ Bx for 1 ≦ i ≦ n. It gives estimates for the first and second moments of N(α, Δ) over the ensemble S n ( 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 S n ( 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, Δ 1, Δ 2) to the homogenous linear congruence λ x 1 ≡ x 2 (mod B) with bounds | x 1| ≦ Δ 1, | x 2| ≦ Δ 2 on the variables, taken over the set of λ with (λ, B) = 1.