TRANSCENDENTAL NUMBERS AND PERIODS JAMES CARLSON Contents. Introduction.. Diophantine approximation I: upper bounds 2.2. Diophantine approximation II: lower bounds 4.3. Proof of the lower bound 5 2. Periods 6 2.. Periods 6 3. Computable numbers 8 4. References. Introduction These are notes for a talk given at Colorado State University on February 2, 205. The numbers we are familiar with fall into a hierarchy of sets of ever greater scope: N Z Q Q C These are the natural numbers, the integers, the rationals, the field of algebraic numbers, and complex numbers, respectively. The field Q is the set of all possible solutions of polynomial equations with rational coefficients. Such equations can be enumerated, so Q, like the field of rational numbers itself, is countable. Nonetheless, the Q is considerably larger than Q in the sense that it infinite dimensional rational vector space. For example, Date: 02-9-204.
2 JAMES CARLSON the numbers p, where p is prime, are linearly independent. These reults have their roots in the Greek mathematics of around 500 BC. Consider now the field of complex numbers. By the work of Georg Cantor, it is an uncountable field a remarkable fact that rests on the Cantor s famous diagonal argument. It follows that most numbers are transcendental. That is, they satisfy no polynomial equation with rational coefficients. Consequently, there is a decomposition C = Q (C Q) where the first term is countable and the second is uncountable. Note also that the first is of measure zero while the second is of infinite measure. The same considerations apply to the deomposition [0, ] = ( Q [0, ]) (C Q) [0, ], except that [0, ] and the second term on the right have measure one. We are now faced with a paradox. The set of algebraic numbers is small both in cardinality and measure relative to the set of all complex numbers. A number chosen at random is transcendental with probability one. Despite this fact, finding explicit transcendental numbers is quite hard. Specific transcendental numbers were exhibted first exhibited in 844 by Liouville. Theorem. The Liouville number is transcendental. In what follows we discuss λ = n=0 0 n!. () The proof of Theorem (2) The definition and properties of a countable class of transcendental numbers, the periods (3) The Konsevich-Zagier philosophy that almost all known transcendental numbers are periods. (4) Yoshinaga s solution of the problem of exhibiting an explicit non-period, analogous to Liouville s construction of λ. (5) The K-Z conjecture that e is not a period. The first item depends on arguments in Diophantine approximation,which we take up now.
TRANSCENDENTAL NUMBERS AND PERIODS 3.. Diophantine approximation I: upper bounds. The first numbers known to be transcendental are the so-called Liouville numbers. To understand them, we must consider the task of approximating an arbitrary real number by rational numbers. If we divide the number line at the rational numbers with denominator q, we divide it into intervals of length /q. Consequently, for any real number α and any integer q > 0, there is an integer p such that α p q < 2q. It turns out, however, that there are specially good ways of approximating a real number by a rational number. These are the continued fractions. Suppose given a real number α > 0. Write it as the sum of an integer and a fractional part, where 0 < α <. Thus for α = π we have α = [α] + { α }, π = 3 + 0.459265.. Let α = /{ α }, and treat α in the same way as we treated α: In the case of π, α = [α ] + { α }, α = 7 + 0.06253484986686 Putting together what we have done s far gives us π = 3 + 7 + 0.06253484986686... This is the beginning of the continued fraction expansion of π. You are urged to work out more terms. The next step is + where ϵ = 0.9.... π = 3 + 7 + 5+ϵ
4 JAMES CARLSON Convergents. Truncate the continued fraction expansion at any stage, discarding the fractional part if necessary. The resulting compound fractions are called convergents, and can be notated as c 0, c, etc. They satisfy In the case of π, c 0 = 3, and lim c n = α n c = 22 7. For the second convergent, set ϵ = 0 to get then c 2 = 3 + 7 +, 5 c = 3 + 7.06666666666666667, and finally + c = 3.45094339622645 Thus π c 2 < 8.4 0 5. By contrast, /5 < 7 0 2. These popular ancient values of π give especially good approximations, e.g., π 22 7 Indeed, we have the following general result: = π 3.428574285744.. < 0.00264493 << 2 7 The convergents p/q of a real number α satisfy α p q < q 2.
TRANSCENDENTAL NUMBERS AND PERIODS 5.2. Diophantine approximation II: lower bounds. The key to showing that λ is transcendental is the following lower bound on approximation of algebraic numbers by rational numbers: Theorem 2. Let α is an irrational number which is a root of an a polynomial of degree n with integer coefficients. Then there is a constant c(α) such that α p q > c(α) q n Since the partial sums of the series in Theorem converge exponentailly fast, we conclude the following: Corollary. Liouville s number λ is transcendental. Much more sophisticated arguments in Diophantine approximation soon came. Theorem 3. (Hermite 873) e is transcendental. Theorem 4. (Lindemann 882) π is transcendental. Theorem 5. (Weierstrass-Lindemann) The exponential of a nonzero algebraic number is transcendental. The transcendence of e is an immediate consequence of the Weierstrass-Lindemann theorem. That theorem also implies the transcendence of π: observe that e 2π = If π is algebraic, then so is 2π. Therefore the expression above is transcendental, a contradiction..3. Proof of the lower bound. Let α be a simple root of a polynomial f(x) with integer coefficients. By the mean value theorem, We can rewrite this as ( ) ( ) ( ) p f(α) f = p q f = p q f a p q q a p q = f ( p q ) ( ) p f q
6 JAMES CARLSON The interval a p q is bounded. Since α is a simple root of f, the derivative of f is nonzero near α and so is bounded below by some non-zero constant c(α). Thus we have Now write where a p q c(α) f f ( ) p = a q q n a = q n f ( ) p q ( p q ) is a a polynomial with integer coefficients evaluated on an integer. Since a 0, a. This observation yields the result. 2. Periods Kontsevich and Zagier posit that almost all transcendental numbers we know are in fact members of a countable set P, the periods. Let us take as a provisional definition of period the length, area, or n-dimensional volume of a set defined by polynomial inequalities with rational coefficients. This set is clearly countable, and it includes where p and q are integers, the roots p p/q q = dx, 0 n a = { 0 x n a } and all algebraic numbers. It also includes the special transcendental numbers dx, log p q = p/q dx x
TRANSCENDENTAL NUMBERS AND PERIODS 7 π = Thus our hierarchy of numbers now reads x 2 +y 2 dx dy N Z Q Q P C Remark. Let a be real number in the interval (0, ) that has infinitiely many nonzero binary digits a i. The Liouville number λ(a) = n= a n 0 n! is transcendental. Thus there is an uncountable set of explicit transcendental numbers which are not periods. 2.. Periods. in algebraic geometry A seemingly much larger class of interesting numbers comes from algebraic geometry. Let X be an algebraic variety defined over the rational numbers, and let ω be any rational differential k-form on X with rational coefficients. Let γ be a homology cycle of dimension k. Then we an form the integral α = This gives a seemingly much more general class of periods. However, Kontsevich and Zagier remark that γ can be approximated by a homologous cycle whose pieces are of the first kind in suitable coordinates. Thus there is no real gain in generality. As an example, consider the form ω = dz/z on the punctured complex line C = { z C z 0 }. Let γ be the unit circle centered a the origin taken with some parameterization. Then γ ω. γ dz z = 2π is a period in the second sense. Replacing γ by a polygon which encircles the origin, we see that it is a period in the seemingly more restricted sense.
8 JAMES CARLSON For another example, consider the elliptic curve E defined by the equation y 2 = x(x )(x a) A homology basis { δ, γ } for E is given in Homology basis A. Figure. Homology basis A The cycle δ encircles the branch points 0 and, winding once around the branch cut from 0 to. The cycle γ starts from a point p on a branch cut from a to infinity, travels in one sheet of E to a point q on the he branch cut, and from there back p in the other sheet. By deforming contours, one sees that γ ω = Such integrals, classically known as elliptic integral, are transcendental the result of deep and difficult theorems of Siegel, Schneider, and Gelfond. a Remark. The periods of an elliptic curve defined over Q generates a field K per called the field of periods. This field has transcendence degree at most four and at least two. For the elliptic curves that have complex multiplication a kind of symmetry which generic elliptic curves do not have, the transcendence degree is exactly two. For more on this topic, see the exposition at Special periods of cubic surfaces. dx y.
TRANSCENDENTAL NUMBERS AND PERIODS 9 Figure 2. Homology basis B 3. Computable numbers A real number is computable if the n-th digit of its decimal expansion is the contents of the n-the square on a tape produced by a Turing machine. Grouping squares into pairs, one has a definition of computability for complex numbers. Let us denote the class of computable numbers by C. Since the set of Turing machines is countable, so is the set C. An alternative definition of computability the following definitions. Definition. A computable function f : N N is a function given by a Turing machine that produces output f(n) on input n. Definition 2. A real number ρ is computable if there are computable functions a(n) and b(n) such that Of course we have ρ a(n) b(n) + < n + lim n but Definition 2 makes a stronger statement. Theorem 6. Periods are computable a(n) b(n) + = ρ, Sketch of proof. Let A be a subset of R n defined by polynomial inequalities with rational coefficients. Let B be a box defined by inequalities b x i b. Let B[n] be the set of
0 JAMES CARLSON subcubes of B[n] of side length /n with vertices of the form a/n, where a is an integer. Let B [n] be the set of subcubes which have a vertex in A. This is is a computable set, since one can determine membership be evaluating polynomial functions with rational coefficients on vertices. Consequently the volume of this set for each n is given by a computable function. Therefore the volume of A is the limit of values a(n)/(b(n) + ), where a and b are computable. To prove computability, however, we must show that the estimate of Definition 2 is satisfied. To this end, note that the volume of the small cubes is O((/n) d ), where A R d. The key point is that the boundary of A is not wild, so that it is covered by O(n d ) small cubes. Thus the error is O(/n). Given this result, our hierarchy is N Z Q Q P C C Example. The number e is computable since we may devise a program for computing the n-th digit from computations of partial sums of the infinite series e = +! + 2! + 3! + Theorem 7. (Yoshinaga) There is a computable number which is not a period. ψ = 0.388832227738246425624300958... Yoshinaga s proof relies yet another set, the elementary computable numbers, EC. He shows that the hierarchy is N Z Q Q P EC C C Using an explicit enumeration of the elementary computable numbers and a knd of Cantor diagonal argument, he exhibits a computable number which is not elementary computable and which is therefore not a period. Yoshinaga defines the elementary computable numbers by defining a class of elementary computable functions. To this end, define rhe operation (x y) to be x y if x y 0 and 0 otherwise. Define bounded summation by by BS : { f : N d N } { g : N d N }
TRANSCENDENTAL NUMBERS AND PERIODS If f(x) is a function of one variable, then x d BSf(x,..., x d ) = f(x,..., x d, k) k=0 BSf(n) = f(0) + f() + + f(n) is a kind of discrete indefinite integral. Bounded product is defined in the same way: BP f(x,..., x d ) = x d k=0 f(x,..., x d, k) Definition 3. The set of elementary computable functions is the smallest set which (a) contains the zero function, the successor functions, and the functions (x, y) x + y, (x, y) (x y) 0, (x, y) xy, and which (b) is closed under composition, bounded summation, and bounded product of functions. Yoshinaga then shows Sets defined by rational polynomial inequalities are elementary computable. The elementary computable functions have an explicit enumeration. By a Cantor diagonal argument, there is a computable number which is not elementary computable. () History: Wikipedia (2) Kontsevich and Zagier 4. References