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

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.

Suppose we want to approximate a number $\alpha$ by a rational $r=\frac{a}{b}$. We obviously care about how close $\alpha$ and $r$ can get – that's the whole point of approximations. We also care about how complex our fraction $r$ is: of course we could approximate $\pi$ by $\frac{3141593}{1000000}$, but since $\frac{355}{113}$ has about the same precision and is much "simpler", we could argue that it is a much better approximation. It just so happens that the size of $b$ is quite a good indicator of both of these intuitive notions, of "closeness" and "complexity". Let me elaborate.

By itself, the denominator is a terrible measure of how close a rational approximation is. Both $\frac{335}{113}$ and something like $\frac{10^{100}}{113}$ have the same denominator, but only one of them even tries to approximate $\pi$. However, let's say that we're trying to approximate $\alpha$ by a rational number, and $b$ is a given. How close can we get? Turns out that if we create intervals of length $\frac{1}{b}$, each centered at a different fraction of the form $\frac{a}{b}$, we can cover the whole number line. Then, $\alpha$ will be in one of these intervals, and at distance of at most $\frac{1}{2b}$ from its center. Here's an example with $b=3$.

Therefore, with a big enough denominator, we can create arbitrarily precise approximations, creating the correlation between the denominator and the notion of closeness that we were talking about earlier.

And what about complexity? This one's easier to justify. If not with the denominator, how else would we quantify how complex a fraction is? The only other reasonable method that comes to mind would be to use the numerator. However, this measure would bias against big numbers, disqualifying them as complex. And although this might correspond with our numerical intuition ($7$ definitely feels simpler than $6198016$), it doesn't correspond with what we're usually interested in when talking about approximations. There's a reason we usually separate the first number in a continued fraction expansion: we mostly don't care for it.

Depending on the context, you might find better indicators of these notions. For example, when talking about intervals in music theory, since we hear frequencies logarithmically, denominators don't really manage to capture that sense of closeness. In this particular example, we also rarely care for fractions too complicated, since past a certain point, our ears are unable to detect the fineness. But in the general case, the denominator is so important, precisely because it captures intuitive concepts within a readily available value.

The denominator tells you whether you're measuring distance in miles or inches. If you approximate something by saying it's six miles and that comes within a quarter inch of being correct, that's impressive. But if you approximate something by saying it's some integer number of inches it it comes within a quarter inch, that's not impressive. How impressive it is depends on how close it comes by comparison to the unit of measurement, i.e. to the denominator.