Let $f(x,y)\in\mathbb{Z}[x,y]$ be a binary cubic form with nonzero discriminant, and for a positive integer $m$ consider the integral representations $f(x,y)=m$. Assume that the number of representations is large, then is it true that a positive proportion of the corresponding lattice points $(x,y)$ lies in the first quadrant (that is, $x$ and $y$ are positive)? More generally, are these lattice points equidistributed by angle (asymptotically, as $m\to\infty$)?
This question is motivated by the paper of Silverman (JLMS (2) 28 (1983), 1-7), which in turn is motivated by the paper of Mahler (PLMS (2) 39 (1935), 431-466). In particular, I wonder if the main theorem in Silverman's paper holds true when we restrict $(x,y)$ to a given angular sector.
Added. Silverman's theorem states that if $f(x,y)=m_0$ defines an elliptic curve over $\mathbb{Q}$ with rank $r$ (for some $m_0$), then the number of integral representations $f(x,y)=m$ is $\gg(\log m)^\frac{r}{r+2}$. I wonder if the same is true for the number of positive integral representations.