Simultaneous Diophantine approximation with a divisibility condition

In a previous paper ([9]), we studied certain sequences of simultaneous rational approximations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> which present some analogy with the continued fractions. We got results around the Littlewood conjecture by using such approximations. Here we show that these results also hold when we add divisibility conditions.