I just finished teaching a course for high school teachers, where I had occasion to talk about the number line. I hadn’t intended to belabor the point, making the assumption that they’d already seen the results of Cantor, and just recalled some highlights, including the stunner that a random real number has probability 1 of being irrational. But the class started to ask question after question, and I took a side line for a while, going considerably deeper into set theory than I had planned. It seems these remarkable results are rather alien these days, and more’s the pity. Can there be any more striking proof of the ubiquity of irrational numbers than Cantor’s famous — or notorious — diagonal argument “that fails”?

One step up the ladder of pedagogical sophistication is the similarly structured proof that a random real number has probability 1 of being transcendental, i.e. of failing to be a root of any polynomial with rational (or, equivalently, integer) coefficients. Thus, transcendental numbers are “everywhere,” while algebraic ones are “nowhere,” even as they are the playground for so much of mathematics, including that jewel in the mathematical crown, algebraic number theory. Despite their beauty, even with algebraic numbers we’re still dealing with a countable set, whence a set of measure zero.

So most numbers are transcendental, and just as there is algebraic number theory, there’s transcendental number theory — and it is notoriously hard. Merely as a sample, although we know that \(e\) and \(\pi\) are transcendental (established by respectively Hermite and Lindemann), we have no proof of the transcendence of their sum: I’d bet the proverbial farm on that contingency, but this is not physics, is it? Where’s the proof?

On the other hand, there are beautiful things about transcendental numbers that we can prove. One jewel is the theorem of A. O. Gelfond and T. Scheider, also connected with Carl Ludwig Siegel (as discussed evocatively in Reid’s *Hilbert*): \(2^{\sqrt2}\) is transcendental — and, more generally, so is \(\alpha^\beta\), whenever \(\alpha\) is a rational other than \(0\) or \(1\) and \(\beta\) is an algebraic irrational. Another gem is the fact that Liouville’s number, \[0.1010010001000010000010000001\dots,\] is transcendental. And so is \(\displaystyle\sum_0^\infty 10^{-n!}\). This is a special case of a more general result, *viz.* that if for a real \(\alpha\) one has that for all \(\mu>0\) the inequality \[0<\left|\alpha-\frac{p}{q}\right|<q^{-\mu}\] has infinitely many solutions, then \(\alpha\) is transcendental. A version of this very beautiful result, due to Liouville, can be found on p. 172 of the book under review.

The preceding theorem is an example of Diophantine approximation, the major focus of the book under review, and the authors start off their Introduction this way: “Diophantine approximation may be roughly described as the branch of number theory concerned with approximations by rational numbers.” Accordingly the natural question to ask in this subject reads something like this: given a number or set of numbers of a particular sort, what is the best approximation we can get by rationals, again of a certain sort — we’re not talking about the obvious elementary result that the rationals are dense in the reals, for example. Indeed, the gold standard for this kind of result the book is the theorem of Thue, Siegel, and Roth, strengthening Liouville’s result. If \(\alpha\) is an irrational algebraic number then for every \(\delta>0\) there are only a finite number of solutions \(\frac{p}{q}\) (in lowest terms)to the inequality \[\left|\alpha-\frac{p}{q}\right|<q^{-(2+\delta)}\]; to get at transcendental numbers, go to the contrapositive. In the present book this result is given on p. 19, with credit to K. F. Roth (1955). The authors characterize this beautiful theorem as the culmination of what Liouville had started and Thue and Siegel had continued.

As the title of the book indicates, however, Corvaja and Zannier are also interested in Diophantine approximation to integral points on algebraic varieties, but before we illustrate this topic, we need an observation. Specifically, it turns out that results along the lines of the Thue-Siegel-Roth Theorem “imply the finiteness of integral points on [certain] families of curves” (cf. p. 60 of the book under review), and this is the direction we now go. Here is Siegel’s Theorem (*loc. cit.*):

“Let \(C\) be an affine irreducible algebraic curve. Suppose that \(C(\mathcal{O}_S)\) is infinite. Then \(C\) has genus zero and at most two points at infinity.

Here \(S\) is a set of places of the underlying number field \(k\) containing all the infinite ones, so that \(\mathcal{O}_S\) is the familiar ring of \(S\)-integers of \(k\) and \(C(\mathcal{O}_S)\) is the according set of \(S\)-integral points of \(C\). Note that (again) going to the contrapositive of this statement, we in fact get (as an implication) what Wikipedia calls Siegel’s Theorem on Integral Points: namely that if \(C\) has positive genus, then it has only a finite number of \(k\)-integral points. Note also that these results make one think of the Mordell Conjecture, a.k.a. Faltings’ Theorem, to the effect that any rational curve of genus at least \(2\) has a finite number of *rational* points on it (classically over \(\mathbb{Q}\), but the proof works over any number field).

This should give at least some of the flavor of what is in this book. There is a great deal more, of course, as a glance at the table of contents will indicate. We meet K. F. Roth and Wolfgang Schmidt, as is to be expected, but also Claude Chevalley and André Weil, seeing that the authors go heavily at themes from algebraic geometry, which, of course, makes the voyage even more exciting. Corvaja and Zennier state in their Preface that “the present work does not require and particular prerequisites; actually, certain basic notions will be recalled, so the general level may be considered fairly elementary. The style is somewhere in between a survey and a detailed account.” This description fits the bill, modulo the usual “mathematical maturity” on the part of the reader. It’s really beautiful mathematics!

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.