Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 2F (Topics In Analysis)

The Liouville approximation theorem states that if $\alpha$ is an irrational algebraic number of degree $d\ge 2$, then there is a constant $c(\alpha )>0$ such that every reduced rational $p/q$ satisfies

Let $P\in \mathbb{Z}[X]$ be the minimal polynomial of $\alpha$, of degree $d$. Since $P$ is irreducible of degree greater than one, $P(p/q)\ne 0$. Moreover,and henceOn the compact interval $I=[\alpha -1,\alpha +1]$, putIf $p/q\in I$, the mean value theorem gives a point $\xi$ between $\alpha$ and $p/q$ such thatThus $|\alpha -p/q|\ge M^{-1}{q}^{-d}$. If $p/q\notin I$, then $|\alpha -p/q|>1\ge q^{-d}$. Taking anyproves the stated strict inequality.

Because both fractions are reduced and $q>v$, they are distinct. Their difference isThe numerator is a nonzero integer, so its absolute value is at least one. Thereforewhere the strict inequality uses $v<q$. This is the separation of distinct reduced fractions.

Since the positive series converges, $a_n\to 0$. We construct the subsequence recursively. Choose $n(1)$ arbitrarily. Once $n(1)<\cdots <n(k-1)$ have been chosen, write the reduced partial sums asChoose $n(k)>n(k-1)$ so far out thatThis is possible because $a_n\to 0$.

LetFor every fixed $j$, each later choice includes $q_j^{-j}$ in its minimum, so

First, $\alpha$ is irrational. If $\alpha =u/v$ were reduced and rational, the strictly increasing rational sequence $s_j\to \alpha$ would have unbounded denominators $q_j$; only finitely many reduced fractions in a bounded interval have bounded denominator. Choose $j>2$ with $q_j>v$. Part (b) would givecontrary to $|\alpha -s_j|<2^{-j}{q}_j^{-j}<q_j^{-2}$.

If $\alpha$ were algebraic of degree $d$, part (a) would give a constant $c>0$ withFor $j>d$ our construction instead giveswhich contradicts the lower bound once $2^{-j}<c$. Hence $\alpha$ is transcendental. This proves the transcendental subseries of a positive rational convergent series construction.