Littlewood's conjecture is defined as follows:
For any real number $x$, define $f(x)$ as the distance between $x$ and the integer nearest to $x$, or $f(x) = \min(x - \lfloor x \rfloor, \lceil x \rceil - x)$.
The statement of the conjecture is that for any two real numbers $\alpha$, $\beta$, $$\liminf_{n \rightarrow \infty} \ n \cdot f(n\alpha) \cdot f(n\beta) = 0$$ taking the $\liminf$ for positive integer values of $n$.
Since the conjecture uses a pair of numbers, it would make sense to consider a "simpler" version of the conjecture, using a single real number only. However, I couldn't find anything on that matter.
For all real number $\alpha$, does $\liminf_{n \rightarrow \infty} \ n \cdot f(n\alpha) = 0?$, Again, we take the $\liminf$ for positive integer values of $n$.
According to the Wikipedia article, Borel(1909) showed that the number of pairs $(\alpha, \beta)$ violating the $2$-variable version is a set of measure zero, and the $2$-variable version connects to future conjectures in group theory. Therefore, I have some questions:
- Is there any proof / counterexample / discussion of my $1$-variable version?
- Are there further connections and/or implications to other problems for the $1$-variable version?
- What about generalisations to $n$-tuples of real values in $\mathbb{R}^n$?
Thanks for your help!
If we consider three or more multipliers f, then we have to consider increasing power of $n$: $$ n^{k}f(n\alpha)f(n\beta)f(n\gamma)\qquad (*) $$ Otherwise, if Littlewood conjecture is valid and still $k=1$, then the limit of the above product is zero thanks to the boundedness of $f(n\gamma)$.