Adamczewski Boris & Yann Bugeaud

On the Littlewood conjecture in simultaneous Diophantine approximation

For any given $\alpha$ with bounded partial quotients, we give an explicit construction of continuum many independent real numbers $\beta$ with bounded partial quotients for which the pair $(\alpha,\beta)$ satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.