Integers expressible as the sum of two rational cubes

We prove that a positive proportion of integers are expressible as the sum of two rational cubes, and a positive proportion are not so expressible, thus proving a conjecture of Davenport. More generally, we prove that a positive proportion (in fact, at least one sixth) of elliptic curves in any cubic twist family have rank 0, and a positive proportion (in fact, at least one sixth) of elliptic curves with good reduction at 2 in any cubic twist family have rank 1. Our method involves proving that the average size of the 2-Selmer group of elliptic curves in any cubic twist family, having any given root number, is 3. We accomplish this by generalizing a parametrization, due to the second author and Ho, of elliptic curves with extra structure by pairs of binary cubic forms. We then use a novel combination of geometry-of-numbers methods and the circle method that builds on earlier work of Ruth and the first author. In particular, we make use of a new interpretation of the singular integral and series arising in the circle method in terms of real and $p$-adic Haar measures on the relevant group. We prove a uniformity estimate for integral points on the relevant quadric, which along with a sieve allows us to prove that the average size of the 2-Selmer group over the cubic twist family is 3. By suitably partitioning the subset of curves in the family with given root number, we effect a further sieve to show that the root number is equidistributed and that the same average, now taken over only those curves of given root number, is again 3. Finally, we apply the $p$-parity theorem of Dokchitser-Dokchitser and a $p$-converse theorem of Burungale-Skinner to conclude. We also prove the analogue of the above results for the sequence of square numbers: namely, we prove that a positive proportion of square integers are expressible as the sum of two rational cubes, and a positive proportion are not.