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The Littlewood conjecture

The Littlewood Conjecture, proposed by UK mathematician John Littewood in 1930, states that for any two real numbers α and β,

$${\displaystyle \liminf _{n\to \infty }\ n\,\Vert n\alpha \Vert \,\Vert n\beta \Vert =0,}$$

where $${\displaystyle \Vert x\Vert :=\min(|x-\lfloor x\rfloor |,|x-\lceil x\rceil |)}$$ is the distance to the nearest integer.

In plain language :

Any two real numbers α and β can be simultaneously approximated at least moderately well by rationals having the same denominator.

As yet, it remains unproved (though there are some partial solutions).

See : Wolfram Mathworld


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