Editing Transcendental number theory (section)

==Mahler's classification==
[[Kurt Mahler]] in 1932 partitioned the transcendental numbers into 3 classes, called '''S''', '''T''', and '''U'''.<ref name=Bug250>{{harvnb|Bugeaud|2012|p=250}}.</ref>  Definition of these classes draws on an extension of the idea of a [[Liouville number]] (cited above).

===Measure of irrationality of a real number===
{{Main|Irrationality measure}}
One way to define a Liouville number is to consider how small a given [[real number]] '''x''' makes linear polynomials |''qx''&nbsp;−&nbsp;''p''| without making them exactly&nbsp;0.  Here ''p'', ''q'' are integers with |''p''|, |''q''| bounded by a positive integer&nbsp;''H''.

Let <math>m(x, 1, H)</math> be the minimum non-zero absolute value these polynomials take and take:

:<math>\omega(x, 1, H) = -\frac{\log m(x, 1, H)}{\log H}</math>
:<math>\omega(x, 1) = \limsup_{H\to\infty}\, \omega(x,1,H).</math>

ω(''x'',&nbsp;1) is often called the '''measure of irrationality''' of a real number&nbsp;''x''. For rational numbers, ω(''x'',&nbsp;1) = 0 and is at least 1 for irrational real numbers. A Liouville number is defined to have infinite measure of irrationality. [[Roth's theorem]] says that irrational real algebraic numbers have measure of irrationality 1.

===Measure of transcendence of a complex number===
Next consider the values of polynomials at a complex number ''x'', when these polynomials have integer coefficients, degree at most ''n'', and [[Height of a polynomial|height]] at most ''H'', with ''n'', ''H'' being positive integers.

Let <math>m(x, n, H)</math> be the minimum non-zero absolute value such polynomials take at <math>x</math> and take:
:<math>\omega(x, n, H) = -\frac{\log m(x, n, H)}{n\log H}</math>
:<math>\omega(x, n) = \limsup_{H\to\infty}\, \omega(x,n,H).</math>

Suppose this is infinite for some minimum positive integer&nbsp;''n''.  A complex number ''x'' in this case is called a '''U&nbsp;number''' of degree&nbsp;''n''.

Now we can define 
:<math>\omega (x) = \limsup_{n\to\infty}\, \omega(x,n).</math> 
ω(''x'') is often called the '''measure of transcendence''' of&nbsp;''x''. If the ω(''x'', ''n'') are bounded, then ω(''x'') is finite, and ''x'' is called an '''S number'''.  If the ω(''x'', ''n'') are finite but unbounded, ''x'' is called a '''T number'''. ''x''&nbsp;is algebraic if and only if&nbsp;ω(''x'')&nbsp;=&nbsp;0.

Clearly the Liouville numbers are a subset of the U numbers.  [[William LeVeque]] in 1953 constructed U numbers of any desired degree.<ref name=LV172>{{harvnb|LeVeque|2002|p=II:172}}.</ref> The [[Liouville numbers]] and hence the U numbers are uncountable sets. They are sets of measure 0.<ref>{{harvnb|Burger|Tubbs|2004|p=170}}.</ref>

T numbers also comprise a set of measure 0.<ref>{{harvnb|Burger|Tubbs|2004|p=172}}.</ref> It took about 35 years to show their existence. [[Wolfgang M. Schmidt]] in 1968 showed that examples exist.  However, [[almost all]] complex numbers are S numbers.<ref name=Bug251/> Mahler proved that the exponential function sends all non-zero algebraic numbers to S numbers:<ref>{{harvnb|LeVeque|2002|pp=II:174–186}}.</ref><ref>{{harvnb|Burger|Tubbs|2004|p=182}}.</ref> this shows that ''e'' is an S number and gives a proof of the transcendence of {{pi}}.  This number {{pi}} is known not to be a U number.{{sfn|Baker|1975|p=86}}  Many other transcendental numbers remain unclassified.

Two numbers ''x'', ''y'' are called '''algebraically dependent''' if there is a non-zero polynomial ''P'' in two indeterminates with integer coefficients such that ''P''(''x'',&nbsp;''y'')&nbsp;=&nbsp;0. There is a powerful theorem that two complex numbers that are algebraically dependent belong to the same Mahler class.<ref name=LV172/><ref>{{harvnb|Burger|Tubbs|2004|p=163}}.</ref>  This allows construction of new transcendental numbers, such as the sum of a Liouville number with ''e'' or&nbsp;{{pi}}.

The symbol S probably stood for the name of Mahler's teacher [[Carl Ludwig Siegel]], and T and U are just the next two letters.

===Koksma's equivalent classification===
[[Jurjen Koksma]] in 1939 proposed another classification based on approximation by algebraic numbers.<ref name=Bug250/><ref name="Baker, p. 87">{{harvnb|Baker|1975|p=87}}.</ref>

Consider the approximation of a complex number ''x'' by algebraic numbers of degree ≤&thinsp;''n'' and height ≤&thinsp;''H''. Let α be an algebraic number of this finite set such that |''x''&nbsp;−&nbsp;α| has the minimum positive value. Define ω*(''x'', ''H'', ''n'') and ω*(''x'', ''n'') by:

:<math>|x-\alpha| = H^{-n\omega^*(x,H,n)-1}.</math>
:<math>\omega^*(x,n) = \limsup_{H\to\infty}\, \omega^*(x,n,H).</math>

If for a smallest positive integer ''n'', ω*(''x'', ''n'') is infinite, ''x'' is called a '''U*-number''' of degree&nbsp;''n''.

If the ω*(''x'', ''n'') are bounded and do not converge to 0, ''x'' is called an '''S*-number''',

A number ''x'' is called an '''A*-number''' if the ω*(''x'', ''n'') converge to&nbsp;0.

If the ω*(''x'', ''n'') are all finite but unbounded, ''x'' is called a '''T*-number''',

Koksma's and Mahler's classifications are equivalent in that they divide the transcendental numbers into the same classes.<ref name="Baker, p. 87"/> The ''A*''-numbers are the algebraic numbers.<ref name=Bug251>{{harvnb|Bugeaud|2012|p=251}}.</ref>

===LeVeque's construction===

Let

:<math>\lambda= \tfrac{1}{3} + \sum_{k=1}^\infty 10^{-k!}.</math>

It can be shown that the ''n''th root of λ (a Liouville number) is a U-number of degree ''n''.<ref>{{harvnb|Baker|1975|p=90}}.</ref>

This construction can be improved to create an uncountable family of U-numbers of degree ''n''. Let ''Z'' be the set consisting of every other power of 10 in the series above for λ. The set of all subsets of ''Z'' is uncountable. Deleting any of the subsets of ''Z'' from the series for λ creates uncountably many distinct Liouville numbers, whose ''n''th roots are U-numbers of degree ''n''.

===Type===
The [[supremum]] of the sequence {ω(''x'',&thinsp;''n'')} is called the '''type'''. Almost all real numbers are S numbers of type 1, which is minimal for real S numbers. Almost all complex numbers are S numbers of type 1/2, which is also minimal. The claims of almost all numbers were conjectured by Mahler and in 1965 proved by Vladimir Sprindzhuk.<ref name="Baker, p. 86">{{harvnb|Baker|1975|p=86}}.</ref>