THE QUALITY OF THE DIOPHANTINE APPROXIMATIONS FOUND BY THE JACOBI-PERRON ALGORITHM AND RELATED ALGORITHMS

Given a vector of real numbers theta = (theta1,..., theta(d)) is-an-element-of R(d), the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d + 1) x (d + 1) convergent matrices {C(n)(theta):n greater-than-or-equal-to 1} whose rows provide Diophantine approximations to theta. Such algorithms A = (T, A) are specified by two maps T:[0, 1]d --> [0, 1]d and A:[0, 1]d --> GL(d + 1, Z), which compute convergent matrices C(n)(theta):=A(T(n)(theta))... A(T(theta))A(theta). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent eta(A)(theta) is the upper bound of those values of delta for which there is some row of the convergent matrices such that for infinitely many values of n that row of C(n)(theta) has SIGMA(i = 1)d\theta(i) - p(i)/q\ less-than-or-equal-to q(-delta). The uniform approximation exponent eta(A)*(theta) is the upper bound of those values of delta such that for all sufficiently large values of n and all rows of C(n)(theta) one has SIGMA(i = 1)d\theta(i) - p(i)/q(i)\ less-than-or-equal-to q(-delta). The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms eta(A)(theta) and eta(A)*(theta) take constant values c(A) and c*(A) on a set of Lebesgue measure one. It establishes the formula c*(A) = 1 - lambda2(A)/lambda1(A) where lambda1(A), lambda2(A) are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A, dmu), where dmu is a T-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that c(A) = c*(A) holds for a large class of such algorithms. These results apply to the d-dimensional Jacobi-Perron algorithm J(d) and Selmer's algorithm. We show that c*(J(2)) greater-than-or-equal-to 1; experimental evidence of Baldwin (1992) indicates (nonrigorously) that c*(J(2)) = 1.374 +/- 0.002. We conjecture that 1 < c*(J(d)) < 1 + 1/d holds for all d greater-than-or-equal-to 2.