Roth's theorems

A few days ago,  The Scotsman published a paper about Klaus Roth's legacy, explaining how he donated his fortune (1 million pounds) to various charities. This paper was reported by some friends on Facebook. Yuri Bilu added the mention that he knew two important theorems of Roth, and since one of them did not immediately reached my mind, I decided to write this post.

The first theorem was a 1935 conjecture of Erdős and Turán concerning arithmetic progression of length 3 that Roth proved in 1952. That is, one is given a set $A$ of positive integers and one seeks for triples $(a,b,c)$ of distinct elements of $A$ such that $a+c=2b$; Roth proved that infinitely many such triples exist as soon as the upper density of $A$ is positive, that is:
\[ \limsup_{x\to+\infty} \frac{\mathop{\rm Card}(A\cap [0;x])}x >0. \]
In 1975, Endre Szemerédi proved that such sets of integers contain (finite) arithmetic progressions of arbitrarily large length. Other proofs have been given by Hillel Furstenberg (using ergodic theory) and Tim Gowers (by Fourier/combinatorical methods); Roth had used Hardy-Littlewood's circle method.

In 1976, Erdős strengthened his initial conjecture with Turán and predicted that arithmetic progressions of arbitrarily large length exist in $A$ as soon as
\[ \sum_{a\in A} \frac 1a =+\infty.\]
Such a result is still a conjecture, even for arithmetic progressions of length $3$, but a remarkable particular case has been proved by Ben Green and Terry Tao in 2004, when $A$ is the set of all prime numbers.

Outstanding as these results are (Tao has been given the Fields medal in 2006 and Szemerédi the Abel prize in 2012), the second theorem of Roth was proved in 1955 and was certainly the main reason for awarding him the Fields medal in 1958. Indeed, Roth gave a definitive answer to a long standing question in diophantine approximation that originated from the works of Joseph Liouville (1844). Given a real number $\alpha$, one is interested to rational fractions $p/q$ that are close to $\alpha$, and to the quality of the approximation, namely the exponent $n$ such that $\left| \alpha- \frac pq \right|\leq 1/q^n$. Precisely, the approximation exponent $\kappa(\alpha)$ is the largest lower bound of all real numbers $n$ such that the previous inequality has infinitely many solutions in fractions $p/q$, and Roth's theorem asserts that one has $\kappa(\alpha)=2$ when $\alpha$ is an irrational algebraic number.

One part of this result goes back to Dirichlet, showing that for any irrational number $\alpha$, there exist many good approximations with exponent  $2$. This can be proved using the theory of continued fractions and is also a classical application of Dirichlet's box principle. Take a positive integer $Q$ and consider the $Q+1$ numbers $q\alpha- \lfloor q\alpha\rfloor$ in $[0,1]$, for $0\leq q\leq Q$; two of them must be less that $1/Q$ apart; this furnishes integers $p',p'',q',q''$, with $0\leq q'<q''\leq Q$ such that $\left| (q''\alpha-p'')-(q'\alpha-p')\right|\leq 1/Q$; then set $p=p''-p'$ and $q=q''-q'$; one has $\left| q\alpha -p \right|\leq 1/Q$, hence $\left|\alpha-\frac pq\right|\leq 1/Qq\leq 1/q^2$.

To prove an inequality in the other direction, Liouville's argument was that if $\alpha$ is an irrational root of a nonzero polynomial $P\in\mathbf Z[T]$, then $\kappa(\alpha)\leq\deg(P)$. The proof is now standard: given an approximation $p/q$ of $\alpha$, observe that $q^d P(p/q)$ is a non-zero integer (if, say, $P$ is irreducible), so that $\left| q^d P(p/q)\right|\geq 1$. On the other hand, $P(p/q)\approx (p/q-\alpha) P'(\alpha)$, hence an inequality $\left|\alpha-\frac pq\right|\gg q^{-d}$.

This result has been generalized, first by Axel Thue en 1909 (who proved an inequality $\kappa(\alpha)\leq \frac12 d+1$), then by Carl Ludwig Siegel and Freeman Dyson in 1947 (showing $\kappa(\alpha)\leq 2\sqrt d$ and $\kappa(\alpha)\leq\sqrt{2d}$). While Liouville's result was based in the minimal polynomial of $\alpha$, these generalisations required to involve polynomials in two variables, and the non-vanishing of a quantity such that $q^dP(p/q)$ above was definitely less trivial. Roth's proof made use of polynomials of arbitrarily large degree, and his remarkable achievement was a proof of the required non-vanishing result.

Roth's proof was “elementary”, making use only of polynomials and wronskians. There are today more geometric proofs, such as the one by Hélène Esnault and Eckart Viehweg (1984) or Michael Nakamaye's subsequent proof (1995) which is based on Faltings's product theorem.

What is still missing, however, is the proof of an effective version of Roth's theorem, that would give, given any real number $n>\kappa(\alpha)$, an actual integer $Q$ such that every rational fraction $p/q$ in lowest terms such that $\left|\alpha-\frac pq\right|\leq 1/q^n$ satisfies $q\leq Q$. It seems that this defect lies at the very heart of almost all of the current approaches in diophantine approximations...