ALGEBRAIC APPROXIMATIONS TO LINEAR COMBINATIONS OF POWERS: AN EXTENSION OF RESULTS BY MAHLER AND CORVAJA-ZANNIER

摘要

For every complex number x, let parallel to x parallel to(Z) := min{vertical bar x-m vertical bar : m is an element of Z}. Let K be a number field, let k is an element of N, and let alpha(1),..., alpha(k) be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of theta is an element of (0, 1) such that there are infinitely many tuples (n, q(1),..., q(k)) satisfying parallel to q(1)alpha(n)(1) + center dot center dot center dot + q(k)alpha(n)(k)parallel to(Z) theta(n) where n is an element of N and q(1),..., q(k) is an element of K* have small logarithmic height compared to n. In the special case when q(1),..., q(k) have the form qi = qci for fixed c1,..., ck, our work yields results on algebraic approximations of c(1)alpha(n)(1) + center dot center dot center dot + c(k)alpha(n)(k) of the form m/q with m is an element of Z and q is an element of K * (where q has small logarithmic height compared to n). Various results on linear recurrence sequences also follow as an immediate consequence. The case where k = 1 and q(1) is essentially a rational integer was obtained by Corvaja and Zannier and settled a long-standing question of Mahler. The use of the Subspace Theorem based on work of Corvaja-Zannier, together with several modifications, plays an important role in the proof of our results.