Covering integers by $x^2+d{y}^2$

Ben Green, Kannan Soundararajan

What proportion of integers $n⩽N$ may be expressed as $x^2+d{y}^2$ for some $d⩽\Delta$, with $x,y$ integers? Writing $\Delta$ as $(\log N)^{\log 2} 2^{α\sqrt{\log \log N…