Transcendental number theory

Use of the term transcendental to refer to an object that is not algebraic dates back to the seventeenth century, when Gottfried Leibniz proved that the sine function was not an algebraic function.[1] The question of whether certain classes of numbers could be transcendental dates back to 1748[2] when Euler asserted[3] that the number log_ab was not algebraic for rational numbers a and b provided b is not of the form b = a^c for some rational c.

Euler's assertion was not proved until the twentieth century, but almost a hundred years after his claim Joseph Liouville did manage to prove the existence of numbers that are not algebraic, something that until then had not been known for sure.[4] His original papers on the matter in the 1840s sketched out arguments using continued fractions to construct transcendental numbers. Later, in the 1850s, he gave a necessary condition for a number to be algebraic, and thus a sufficient condition for a number to be transcendental.[5] This transcendence criterion was not strong enough to be necessary too, and indeed it fails to detect that the number e is transcendental. But his work did provide a larger class of transcendental numbers, now known as Liouville numbers in his honour.

Liouville's criterion essentially said that algebraic numbers cannot be very well approximated by rational numbers. So if a number can be very well approximated by rational numbers then it must be transcendental. The exact meaning of "very well approximated" in Liouville's work relates to a certain exponent. He showed that if α is an algebraic number of degree d ≥ 2 and ε is any number greater than zero, then the expression

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{d+\varepsilon }}}}$

can be satisfied by only finitely many rational numbers p/q. Using this as a criterion for transcendence is not trivial, as one must check whether there are infinitely many solutions p/q for every d ≥ 2.

In the twentieth century work by Axel Thue,[6] Carl Siegel,[7] and Klaus Roth[8] reduced the exponent in Liouville's work from d + ε to d/2 + 1 + ε, and finally, in 1955, to 2 + ε. This result, known as the Thue–Siegel–Roth theorem, is ostensibly the best possible, since if the exponent 2 + ε is replaced by just 2 then the result is no longer true. However, Serge Lang conjectured an improvement of Roth's result; in particular he conjectured that q^2+ε in the denominator of the right-hand side could be reduced to ${\displaystyle q^{2}(\log q)^{1+\epsilon }}$.

Roth's work effectively ended the work started by Liouville, and his theorem allowed mathematicians to prove the transcendence of many more numbers, such as the Champernowne constant. The theorem is still not strong enough to detect all transcendental numbers, though, and many famous constants including e and π either are not or are not known to be very well approximable in the above sense.[9]

Fortunately other methods were pioneered in the nineteenth century to deal with the algebraic properties of e, and consequently of π through Euler's identity. This work centred on use of the so-called auxiliary function. These are functions which typically have many zeros at the points under consideration. Here "many zeros" may mean many distinct zeros, or as few as one zero but with a high multiplicity, or even many zeros all with high multiplicity. Charles Hermite used auxiliary functions that approximated the functions ${\displaystyle e^{kx}}$ for each natural number ${\displaystyle k}$ in order to prove the transcendence of ${\displaystyle e}$ in 1873.[10] His work was built upon by Ferdinand von Lindemann in the 1880s[11] in order to prove that e^α is transcendental for nonzero algebraic numbers α. In particular this proved that π is transcendental since e^πi is algebraic, and thus answered in the negative the problem of antiquity as to whether it was possible to square the circle. Karl Weierstrass developed their work yet further and eventually proved the Lindemann–Weierstrass theorem in 1885.[12]

In 1900 David Hilbert posed his famous collection of problems. The seventh of these, and one of the hardest in Hilbert's estimation, asked about the transcendence of numbers of the form a^b where a and b are algebraic, a is not zero or one, and b is irrational. In the 1930s Alexander Gelfond[13] and Theodor Schneider[14] proved that all such numbers were indeed transcendental using a non-explicit auxiliary function whose existence was granted by Siegel's lemma. This result, the Gelfond–Schneider theorem, proved the transcendence of numbers such as e^π and the Gelfond–Schneider constant.

The next big result in this field occurred in the 1960s, when Alan Baker made progress on a problem posed by Gelfond on linear forms in logarithms. Gelfond himself had managed to find a non-trivial lower bound for the quantity

${\displaystyle |\beta _{1}\log \alpha _{1}+\beta _{2}\log \alpha _{2}|\,}$

where all four unknowns are algebraic, the αs being neither zero nor one and the βs being irrational. Finding similar lower bounds for the sum of three or more logarithms had eluded Gelfond, though. The proof of Baker's theorem contained such bounds, solving Gauss' class number problem for class number one in the process. This work won Baker the Fields medal for its uses in solving Diophantine equations. From a purely transcendental number theoretic viewpoint, Baker had proved that if α₁, ..., α_n are algebraic numbers, none of them zero or one, and β₁, ..., β_n are algebraic numbers such that 1, β₁, ..., β_n are linearly independent over the rational numbers, then the number

${\displaystyle \alpha _{1}^{\beta _{1}}\alpha _{2}^{\beta _{2}}\cdots \alpha _{n}^{\beta _{n}}}$

is transcendental.[15]

In the 1870s, Georg Cantor started to develop set theory and, in 1874, published a paper proving that the algebraic numbers could be put in one-to-one correspondence with the set of natural numbers, and thus that the set of transcendental numbers must be uncountable.[16] Later, in 1891, Cantor used his more familiar diagonal argument to prove the same result.[17] While Cantor's result is often quoted as being purely existential and thus unusable for constructing a single transcendental number,[18][19] the proofs in both the aforementioned papers give methods to construct transcendental numbers.[20]

While Cantor used set theory to prove the plenitude of transcendental numbers, a recent development has been the use of model theory in attempts to prove an unsolved problem in transcendental number theory. The problem is to determine the transcendence degree of the field

${\displaystyle K=\mathbb {Q} (x_{1},\ldots ,x_{n},e^{x_{1}},\ldots ,e^{x_{n}})}$

for complex numbers x₁, ..., x_n that are linearly independent over the rational numbers. Stephen Schanuel conjectured that the answer is at least n, but no proof is known. In 2004, though, Boris Zilber published a paper that used model theoretic techniques to create a structure that behaves very much like the complex numbers equipped with the operations of addition, multiplication, and exponentiation. Moreover, in this abstract structure Schanuel's conjecture does indeed hold.[21] Unfortunately it is not yet known that this structure is in fact the same as the complex numbers with the operations mentioned; there could exist some other abstract structure that behaves very similarly to the complex numbers but where Schanuel's conjecture doesn't hold. Zilber did provide several criteria that would prove the structure in question was C, but could not prove the so-called Strong Exponential Closure axiom. The simplest case of this axiom has since been proved,[22] but a proof that it holds in full generality is required to complete the proof of the conjecture.

One way to define a Liouville number is to consider how small a given real number x makes linear polynomials |qx − p| without making them exactly 0. Here p, q are integers with |p|, |q| bounded by a positive integer H.

Let ${\displaystyle m(x,1,H)}$ be the minimum non-zero absolute value these polynomials take and take:

${\displaystyle \omega (x,1,H)=-{\frac {\log m(x,1,H)}{\log H}}}$

${\displaystyle \omega (x,1)=\limsup _{H\to \infty }\,\omega (x,1,H).}$

ω(x, 1) is often called the measure of irrationality of a real number x. For rational numbers, ω(x, 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.

Next consider the values of polynomials at a complex number x, when these polynomials have integer coefficients, degree at most n, and height at most H, with n, H being positive integers.

Let ${\displaystyle m(x,n,H)}$ be the minimum non-zero absolute value such polynomials take at ${\displaystyle x}$ and take:

${\displaystyle \omega (x,n,H)=-{\frac {\log m(x,n,H)}{n\log H}}}$

${\displaystyle \omega (x,n)=\limsup _{H\to \infty }\,\omega (x,n,H).}$

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

Now we can define

${\displaystyle \omega (x)=\limsup _{n\to \infty }\,\omega (x,n).}$

ω(x) is often called the measure of transcendence of 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 is algebraic if and only if ω(x) = 0.

Clearly the Liouville numbers are a subset of the U numbers. William LeVeque in 1953 constructed U numbers of any desired degree.[24] The Liouville numbers and hence the U numbers are uncountable sets. They are sets of measure 0.[25]

T numbers also comprise a set of measure 0.[26] 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.[27] Mahler proved that the exponential function sends all non-zero algebraic numbers to S numbers:[28][29] this shows that e is an S number and gives a proof of the transcendence of π. This number π is known not to be a U number.[30] 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, y) = 0. There is a powerful theorem that two complex numbers that are algebraically dependent belong to the same Mahler class.[24][31] This allows construction of new transcendental numbers, such as the sum of a Liouville number with e or π.

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.

Jurjen Koksma in 1939 proposed another classification based on approximation by algebraic numbers.[23][32]

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

${\displaystyle |x-\alpha |=H^{-n\omega ^{*}(x,H,n)-1}.}$

${\displaystyle \omega ^{*}(x,n)=\limsup _{H\to \infty }\,\omega ^{*}(x,n,H).}$

If for a smallest positive integer n, ω*(x, n) is infinite, x is called a U*-number of degree 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 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.[32] The A*-numbers are the algebraic numbers.[27]

Let

${\displaystyle \lambda ={\tfrac {1}{3}}+\sum _{k=1}^{\infty }10^{-k!}.}$

It can be shown that the nth root of λ (a Liouville number) is a U-number of degree n.[33]

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 nth roots are U-numbers of degree n.

The supremum of the sequence {ω(x, 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.[34]