A Simple Generalization of the Littlewood Conjecture

Question

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.

Answer 1

$ \vert x(r x-y)\vert <1$ implies that $y/x$ is a convergent of the continued fraction expansion of $r$. This can be used to construct a counter-example as follows. Consider for $r$ for example a fairly large irrational quadratic real number with constant continued fraction $[l,l,l,l,\dots]$. Let $d_1 < d_2 < d_3,\dots$ be the sequence of denominators appearing in the convergents of $r$. Consider $s$ of the form $s=1/2\sum_{n=1}^\infty \alpha_n/d_n$ with $\alpha_i\in\{0,1}$ recursively defined such that the distance of $d_i(1/2\sum_{n=1}^i\alpha_i/d_i)$ to the nearest integer is $\geq 1/4$. This implies that the distance of $d_i(1/2\sum_{n=1}^\infty \alpha_n/d_n)$ to the nearest integer is $>1/4-\epsilon$ for $l$ large enough. Indeed, the sequence $d_1,d_2,\dots$ grows roughly like a geometric sequence of argument $l$. This implies that $d_i(1/2\sum_{n=i+1}^\infty \alpha_n/d_n)$ is at most of absolute value roughly given by $1/(2l(1-1/l))=1/(2l-2)$ which can be made arbitrarily small by choosing $l$ large.

Answer 2

This answer pieces together the various comments made by Roland Bacher, SJR and gowers previously. The proposed generalization of Littlewood's conjecture is false.

As suggested by Roland and SJR, take $r$ to be the golden ratio $(1+\sqrt{5})/2$.
Then $x|rx-y| \le 1$ only when $x$ runs over the Fibonacci numbers $F_k$. Now we want to show that there is an irrational number $s$ such that $F_k s$ is bounded away from integers.

As suggested by gowers there do exist such $s$ for any lacunary sequence $n_k$ (that is a sequence with $n_{k+1}/n_k \ge 1+ c>1$ for all large $k$), and so in particular for the Fibonacci numbers. This is related to a conjecture of Erdos, that for any lacunary sequence $n_k$ there exist irrational numbers $\alpha$ with $n_k \alpha$ not being dense $\mod 1$. Erdos's problem was settled independently by de Mathan and Pollington in a stronger form, showing that there exist many such $\alpha$. We need in fact that the values $\mod 1$ are bounded away from $0$. This is worked out in detail using Pollington's argument in a recent nice preprint of Haynes and Munday (see Lemma 1 of http://arxiv.org/abs/1308.0208 ).

Answer 3

I can't find the paper online, but this looks rather like a question that is answered by a paper of Pollington and Vellani. Here is a link to an abstract of the paper. (It may be clear from the abstract that they answer your question -- I am feeling lazy and so have not checked.)

http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=121841