From: israel@math.ubc.ca (Robert Israel)
Newsgroups: sci.math
Subject: Re: conjecture on weird series
Date: 1 Mar 1996 01:49:14 GMT

In article <4h30oc$sv1@nntp.ucs.ubc.ca>, israel@math.ubc.ca (Robert Israel) writes:

|> For every irrational number x there are infinitely many m/n with
|> |x - m/n| < 1/n^2.  I'm not sure about the restriction to n odd, but I
|> believe that can be handled as well.

Yes, W.T. Scott (Bull. Amer. Math. Soc. 46 (1940), 124 - 129) showed that
for every irrational number x there are infinitely many m/n with 
|x - m/n| < 1/n^2 and m/n in any one of the categories odd/odd, even/odd and
odd/even.

-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             (604) 822-3629
University of British Columbia            fax 822-6074
Vancouver, BC, Canada V6T 1Y4

==============================================================================

From: kubo@abel.harvard.edu (Tal Kubo)
Newsgroups: sci.math
Subject: Rational Approximations to Pi (was Re: conjecture on weird series)
Date: 2 Mar 1996 08:40:12 GMT

Robert Israel <israel@math.ubc.ca> wrote:
>  
>  [re: solutions m,n  of  |Pi - m/n| < c/n^q,  for c>0 and n>>0] 
>
> The best bound that has been proven, as far as I know, is that 
> with q=42 and some c there are no solutions (K. Mahler, Indag. 
> Math. 15 1953) 30-42).

This comes up often enough that it may warrant FAQ'ing.

The record irrationality measures to date (ie, smallest known 
replacements for '42' above) are:

  for  Pi:  8.016045...     M. Hata, Acta Arithmetica 63 (1993) 335-349

       Pi^2:  7.81         |  D.V Chudnovsky & G.V Chudnovsky,
                           |  Approximations and Complex Multiplication
       Pi*sqrt(2): 16.67   |   According to Ramanujan,  in _Ramanujan
                           |  Revisited_ (Urbana-Champaign 1987), 375-472
       Pi*sqrt(3): 5.8     |  publ. Academic Press, 1988