
Re: might one improve on Hurwitz' Theorem for Diophantine approximations to pi?
Posted:
Apr 7, 2013 8:28 AM


On Friday, April 5, 2013 1:17:29 AM UTC7, David Bernier wrote:
> Hurwitz' Theorem on Diophantine approximations states that, > if alpha is an irrational number in the reals R,
> then for infinitely many positive integers m,n with > gcd(m, n) = 1, one has: alpha  m/n < 1/(sqrt(5)*n^2) .
> Cf.: > < http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_%28number_theory%29 > .
> Do number theorists contemplate as "somewhat possible" > that for alpha=pi, one might be able to prove a bit > more without a 10+ year effort by many, i.e. > an improvement by epsilon without huge effort?
> The improvement would go like this: > pi  m/n < C/n^2 for infinitely many coprime positive > integers m, n for a stated C (e.g. "C = 1/sqrt(5)  1/10^100." ), > with C < 1/sqrt(5) ...
I believe it's well known that the constant C = 1/sqrt(5) is only required for special numbers closely related to the golden ratio. For all other numbers, a considerably smaller constant applies.

