Diophantine approximation is about how well real numbers can be approximated by rationals. Say I give you a real number $\alpha$, and I ask you to approximate it by a rational number $a/q$, where $q$ is not too large. A naive strategy would be to first choose $q$ arbitrarily, and to then choose the nearest integer $a$ to $q \alpha$. This would give $| \alpha - a/q| \le 1/(2q)$, and $\pi \approx 3.14$. Dirichlet, introducing the pigeonhole principle, showed non-constructively that there are infinitely many solutions to $| \alpha - a/q| \le 1/q^2$, and one can use continued fractions to find such approximations, for instance $\pi \approx 22/7$.
Metric diophantine approximation is about the typical rate of approximation. There are values of $\alpha$, such as the golden ratio, for which one can't do much better than Dirichlet's theorem. However, for all $\alpha$ away from a set of Lebesgue measure zero, one can beat it by a factor of $\log q$ and more. Khintchine's theorem is prototypical, asserting that if $\psi: \mathbb N \to [0, \infty)$ is decreasing then \[ \mathrm{meas} \{ \alpha \in [0,1]: \exists^\infty (q,a) \in \mathbb N \times \mathbb Z \quad | \alpha - a/q| < \psi(q)/q \} = \begin{cases} 1, & \text{if } \sum_{q=1}^\infty \psi(q) = \infty \\ 0,&\text{if } \sum_{q=1}^\infty \psi(q) < \infty. \end{cases} \] One can prove these sorts of results using the Borel-Cantelli lemmas, from probability theory: making a ball of radius $\psi(q)/q$ around each $a/q$, and grouping together the ones with the same $q$, the idea is to show that pairs of groups overlap more or less independently.
According to my mathematical upbringing, all phenomena are explained by the dichotomy between structure and randomness: either there is structure present, or else there is (pseudo)randomness. The probabilistic considerations above had initially led me to believe that randomness was the key to understanding metric diophantine approximation, but after working in the area for a while my opinion is closer to the opposite! The denominators of the good approximations to $\alpha$ lie in Bohr sets (after Harald Bohr, brother of the eminent physicist Niels Bohr) \[ B_N(\alpha, \delta) := \{ n \le N: \| n \alpha \| \le \delta \} \subset \mathbb N, \] where $\| \cdot \|$ denotes distance to the nearest integer. A central tenet of additive combinatorics is that Bohr sets look like generalised arithmetic progressions (GAPs).
I built the GAPs using continued fractions, enabling me to make progress towards the infamous Littlewood (c. 1930) and Duffin-Schaeffer (1941) conjectures. The former is about approximating two numbers at once in a multiplicative sense, that is to find approximations $a/q, b/q$ to $\alpha,\beta$ for which \[ \Bigl | \alpha - \frac a q \Bigr | \cdot \Bigl |\beta - \frac b q \Bigr| < \frac {10^{-100}}{q^3}, \] and the latter is about approximation by reduced fractions. With Niclas Technau, we have since developed a higher-dimensional structural theory using the geometry of numbers. Going forward, I hope to establish a Khintchine-type law for multiplicative approximation on planar curves.
Sam Chow, Oxford Mathematics