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Tim Gowers has a list of possible Polymath projects up, where he mentions the Littlewood Conjecture: This states, informally, that all reals can be approximated reasonably well, in the sense that if we let the denominator of an approximation [tex]\frac{u}{v}[/tex] grow, the error term becomes smaller faster than the denominator becomes larger.

Somehow, I was unaware of this fascinating problem: Cursory examination yields a lot of proof-wise trivial, but still interesting, results. Specifically: If [tex]\lim \inf_{n \rightarrow \infty} \{ n ||n\alpha ||\} = 0[/tex] for all [tex]\alpha[/tex] then the conjecture if obviously true.

So, it seems obvious that the question of whether [tex]\lim \inf_{n \rightarrow \infty} \{ n ||n\alpha ||\} = 0[/tex] for all [tex]\alpha[/tex] is also open. But this seems like it should be easier to understand and prove than the Littlewood conjecture itself, because we only have the single real to contend with.

Hence, my question: Does anyone know of any work on approximating a single real with rationals, where the size of the rational is taken into account? Are there any well-known bounds?

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# Approximating the reals by rationals (Littlewood's Conjecture)

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