The Littlewood Conjecture asserts that for all real numbers $r$ and $s$, and for every
$\epsilon > 0$, the inequality
$|x(rx-y)(sx-z)| < \epsilon$ is solvable in integers $x, y, z$ with
$x > 0$.
The Littlewood conjecture is clearly a consequence of the following:
For all real numbers $r$ and $s$, and for every
$\epsilon > 0$ the inequalities
$|x(rx-y)| < 1, \,\,|sx-z| < \epsilon$ are solvable in integers $x, y, z$ with
$x > 0$.
Does anyone know a counter-example to the latter statement? Does anyone know of any references to it in the literature?
Note that the inequality
$|x(rx-y)| < 1$ always has infinitely many solutions $(x,y)$ with
$x > 0$. This is a consequence of Dirichlet's Approximation Theorem. So it is natural to ask: "How does $sx$ behave mod 1 as $(x,y)$ runs through the solutions of
$|x(rx-y)| < 1$?" For example, can the closure of the $sx$ mod 1 contain some non-empty open set and be disjoint from another?
Experiments with Sage seem to "suggest" that the numbers $sx$ are either dense mod 1 or have just finitely many limit points mod 1, depending on whether the numbers $1,r,s$ are linearly independent over the rationals. Again, any counter-examples or references relevant to this statement would be appreciated.