On the Littlewood conjecture in simultaneous Diophantine approximation

Abstract

For any given real number $\hat{I}\pm$ with bounded partial quotients, we construct explicitly continuum many real numbers ${\hat{I}}^2$ with bounded partial quotients for which the pair $(\hat{I}\pm ,{\hat{I}}^2)$ satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.