V.22 Liouville’s Theorem and Roth’s Theorem

One of the most famous theorems in mathematics is the statement that is irrational. This means that there is no pair of integers p and q such that = p/q, or equivalently that the equation p² = 2q² has no integer solutions apart from the trivial solution p = q = 0. The argument that proves this can be considerably generalized, and, in fact, if P(x) is any polynomial with integer coefficients and leading coefficient 1, then all its roots are either integers or irrational numbers. For example, since x³ + x - 1 is negative when x = 0 and positive when x = 1 it must have a root strictly between 0 and 1. This root is not an integer, so it must be irrational.

Once one has proved that a number is irrational, it may seem as though not much more can be said. However, this is very far from true: given an irrational number, one can ask how close it is to being rational, and fascinating and extremely difficult questions arise as soon as one does so.

It is not immediately obvious what this question means, since every irrational number can be approximated as closely as you like by rational numbers. For example, the decimal expansion of begins 1.414213. . . , which tells us that is within 1/100 000 of the rational number 141 421/100 000. More generally, for any positive integer q we can let p be the largest integer such that p/q < , and then p/q will be within 1/q of . In other words, if we want an approximation to with accuracy 1/q, we can obtain it if we use a denominator of q.

However, we can now ask the following question: are there denominators q for which one can one obtain an accuracy much better than 1/q? The answer turns out to be yes. To see this, let N be a positive integer and consider the numbers 0, , 2,. . . , N. Each of these can be written in the form m + α, where m is an integer and α, the fractional part, lies between 0 and 1. Since there are N + 1 numbers, at least two of their fractional parts must be within 1/N of each other. That is, we can find integers r < s between 0 and N such that if we write r = n+α and s = m+β, then |α-β| ≤ 1/N. Thus, if we set = α - β, we have (s - r) = n - m + and || ≤ 1/N. If we now let q = s - r and p n - m, then = p/q + /q, so | - p/q ≤ 1/qN. Since N ≥ q, 1/qN ≥ q, 1/q², so for at least some positive integers q we can achieve an accuracy of 1/q² using a denominator of q.

A different argument shows that we cannot do substantially better than this. Let p and q be any two positive integers. Since is irrational, p² and 2q² are distinct positive integers, which implies that |p² - 2q²| ≥ 1. On factorizing, we deduce that |p - | (p + q ) ≥ 1. We can now divide through by q² and obtain the inequality |p/q - (p/q + ) ≤ 1/q². We may as well assume that p/q is less than 2, since otherwise it is not a good approximation to . But then p/q + is less than 4, so the inequality implies that p/q - ≤ 1/4q². Thus, with a denominator of q we cannot achieve an accuracy better than 1/4q².

A generalization of this argument proves Liouville’s theorem: if x is an irrational root of a polynomial of degree d and p and q are integers, then |p/q - x| cannot be substantially smaller than 1/q^d. When x = this reduces to what we have just shown, since then x² - 2 = 0 and we can set d = 2. However, from Liouville’s theorem we know many similar facts, such as that |p/q - | cannot be substantially smaller than 1/q³.

Roth’s theorem, proved in 1955, is the astonishing assertion that the power d that appears in Liouville’s theorem can be improved—almost as far as 2. To be precise, given any irrational root x of any polynomial, and any number r > 2, there is a constant c > 0 with the property that |p/q - x| is always at least as big as c/q^r. (The proof gives no information whatsoever about c beyond the fact that it is positive. It is a major open problem to understand something about how c depends on r and x.)

To see why this is a much deeper result than Liouville’s theorem, consider the example of . Underlying the proof that |p/q - | is never much smaller than 1/q³ is the simple fact that p³ and 2q³ are distinct integers and therefore differ by at least 1. In order to prove a substantially better result such as Roth’s theorem, one must show much more: that p³ and 2q³ differ by an amount that grows as p and q grow. For example, if one wishes to prove Roth’s theorem when r = , it is necessary to show that p³ and 2q³ must always differ by an amount comparable to or greater than , and it is far from obvious why this should be so.

The Mordell Conjecture

See RATIONAL POINTS ON CURVES AND THE MQRDELL CONJECTURE [V.29]