The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Abstract

Given a vector of real numbersθ=(θ1,...θd)∈ℝd, the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C(n)(ϕ):n≥1} whose rows provide Diophantine approximations to ϕ. Such algorithms are specified by two mapsT:[0, 1]d→[0, 1]d and A:[0,1]d→GL(d+1,ℤ), which compute convergent matrices C(n)(θ))...A(T(θ))A(θ). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of δ for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C(n)(ϕ) has\(\sum\nolimits_{i = 1}^d {\left| {\theta _i - p_i /q} \right| \leqslant q^{ - \delta } } \). The uniform approximation exponent is the upper bound of those values of δ such that for all sufficiently large values ofn and all rows of C(n)(ϕ) one has\(\sum\nolimits_{i = 1}^d {\left| {\theta _i - p_i /q_i } \right| \leqslant q^{ - \delta } } \). The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,dμ), wheredμ is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that ; experimental evidence of Baldwin (1992) indicates (nonrigorously) that . We conjecture that holds for alld≥2.