From: tchow@my-dejanews.com
Newsgroups: sci.math.research
Subject: Approximating pi by algebraic numbers
Date: Fri, 12 Jun 1998 18:02:22 GMT

There is a sizable literature on the problem, discussed in several recent
articles, of approximating pi by algebraic numbers; it goes under the name
of "Mahler's classification of transcendental numbers."

Kevin Brown's suggested measure of the "goodness of fit" of an algebraic
number is close to the standard one, but not quite the same.  Define the
"height" h of a polynomial equation with integer coefficients to be the
maximum of the absolute values of its coefficients, and let n be its degree.
Then the usual measure is (roughly speaking)

     # decimal digits matched
     ------------------------
        n * (# digits in h)

For fixed n, we can take the lim sup of this as h -> oo to get a number w_n.
Then we can let w be the lim sup of w_n as n -> oo.  Brown's suggestion, if
I understand correctly, is that w is finite.

It is indeed true that w is finite for all numbers except for a set of measure
zero.  It is known that e has this property, but it is not known if pi does
(though it is known that for pi, w_n is finite for all n).

For more information, see chapter 8 of Baker's _Transcendental_Number_Theory_.

Tim Chow      tchow AT alum DOT mit DOT edu

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