On the infinitude of elliptic curves over a number field with prescribed small rank

For any number field $K$ and integer $0\le r\le 4$, we prove that there are infinitely many elliptic curves over $K$ of rank $r$. Our elliptic curves are obtained by specializing well-chosen nonisotrivial elliptic curves over the function field $K(T)$. We use a result of Kai, which generalizes work of Green, Tao and Ziegler to number fields, to choose our specializations so that we have control over the bad primes and can perform a $2$-descent to compute ranks.