Approximating the reals by rationals (Littlewood's Conjecture)

[funkstar]

Hi all,

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?

 

[ramsey2879]

Hi all,

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?

A irrational number is an infinite decimal string but can be approximated more closely than a prior estimate merely by truncating the string after determiining the additional decimal places, i.e. to make the denominator a higher power of 10 than before. Mathematicians have been figuring the ratio of the circumference of a circle to its the radius since B.C. and still a few are not yet satisfied with the number of decimal places calculated by the most modern computers since to calculate pi seems to be used as a bench mark for the power or speed of a computer.

 

[tomtom2357]

lim inf _n→∞{n||nα||}=0 is false. The golden ratio is a counterexample