Intuition behind Diophantine approximation: why do we express the bound as function of denominators

Question

The field $\mathbb Q$ is dense in $\mathbb R$, so we can approximate a real number $\alpha$ by an arbitrarly close rational number $r=\frac{a}{b}$.

---

The purpose of Diophantine approximation is to find $\frac{a}{b}$ approximating $\alpha$ in a way that $|\alpha-\frac{a}{b}|<f(b)$ where $f$ is a function. Roughly speaking we want to compare the "exactness" of the approximation with the denominator of the approximating number. In other words we would like $\frac{a}{b}$ to be both close enough to $\alpha$ but also $b$ should be small (in absolute value).

My question is: why are we so interested in the size of "$b$"? Why such emphasis on denominators? To me it seems a naive starting point for a whole theory. Evidently there is something I don't understand.

Answer 1

Here we take a look at chapter 11: Approximation of irrationals by rationals from the classic An Introduction to the Theory of Numbers by G.H. Hardy and E.M. Wright.

The chapter starts with:

The problem considered in this chapter is that of the approximation of a given number $\xi$, usually irrational, by a rational fraction \begin{align*} r=\frac{p}{q}. \end{align*} We suppose throughout that $0<\xi<1$ and that $p/q$ is irreducible.

From this first sentence we can already see

• we can stick to numbers $\xi$ in the interval $(0,1)$ implying that large numerators are not of interest

• we can put the focus on irrationals, since approximating rationals by rationals is presumably not that challenging (and shown by the authors soon after the beginning of this chapter)

The authors continue as we might expect:

... We ask now how simply or, what is essentially the same thing, how rapidly can we approximate to $\xi$? Given $\xi$ and $\epsilon$, how complex must $p/q$ be (i.e. how large $q$) to secure an approximation with the measure of accuracy $\epsilon$? Given $\xi$ and $q$, or some upper bound for $q$, how small can we make $\epsilon$?

and they state some basic facts like:

... given $\xi$ and $n$, then there exist integers $p,q$ with $0<q\leq n$, so that \begin{align*} \left|\frac{p}{q}-\xi\right|\leq \frac{1}{q(n+1)}<\frac{1}{q^2}\tag{1} \end{align*}

...[followed by]...

(Theorem 185): If $\xi$ is irrational, then there is an infinity of fractions $p/q$ which satisfy (1).

Soon after that the authors go somewhat closer to the essential part of the theory by commenting the question above:

These questions, however, are not the questions most interesting to us now. We are not so much interested in approximations to $\xi$ with an arbitrary denominator $q$, as in approximations with an appropriately selected $q$.

• For example, there is no great interest in approximations to $\pi$ with denominator $11$; what is interesting is that two particular denominators $7$ and $113$, give the very striking approximations $\frac{22}{7}$ and $\frac{335}{113}$.

• We should ask, not how closely we can approximate to $\xi$ with an arbitrary $q$, but how closely we can approximate for an infinity of values of $q$.

• We shall therefore be occupied, throughout the rest of this chapter, with the following problem: for what $\phi=\phi(\xi,q)$, or $\phi=\phi(q)$, is it true, for a given $\xi$, or for all $\xi$, or for all $\xi$ of some interesting class, \begin{align*} \left|\frac{p}{q}-\xi\right|\leq \frac{1}{\phi} \end{align*} has an infinity of solutions.

We see it is not only the emphasis of the denominator $q$ of the approximating fraction, but much more the search for an interesting class of values which is to approximate and the kind of approximation which enables us to detect structural properties of irrational numbers.

One of the interesting approaches defines orders of approximation with the help of the $n$-th convergent $q_n$ of a continued fraction approximating $\xi$.

(section 11.4): We shall say that $\xi$ is approximable by rationals to order $n$ if there is a $K(\xi)$, depending only on $\xi$, for which \begin{align*} \left|\frac{p}{q}-\xi\right|<\frac{K(\xi)}{q_n} \end{align*} has an infinity of solutions.

From this some interesting consequences are:

• (Theorem 186): A rational is approximable to order $1$, and to no higher order.

• (Theorem 187): Any irrational is approximable to order $2$.

Recalling that algebraic numbers satisfying an algebraic equation of degree $n$, but none of lower degree, are called algebraic numbers of degree $n$ the following holds:

• (Theorem 191): A real algebraic number of degree $n$ is not approximable to any order greater than $n$.

The last theorem shows that this theory of approxmation might enable us to reveal information about the transcendence of numbers. Actually, the last section of this chapter has the title The Transcendence of $\pi$, showing us the power and the beauty of this theory.

Hint: To indicate the term classic of the referred book, I'd like to state from the nicely written booklet Irrational numbers by Ivan Niven from the notes at the end of chapter $4$: Approximation by Rationals:

• An excellent exposition is to be found in G. H. Hardy and E. M. Wright, Theory of Numbers, Chapters $11$, $23$, $24$.