Approximation to real numbers by cubic algebraic integers I

In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number \xi by algebraic integers of degree at most three. They did so, using geometry of numbers, by resorting to the dual problem of finding simultaneous approximations to \xi and \xi^2 by rational numbers with the same denominator. In this paper, we show that their measure of approximation for the dual problem is optimal and that it is realized for a countable set of real numbers \xi. We give several properties of these numbers including measures of approximation by rational numbers, by quadratic real numbers and by algebraic integers of degree at most three.