The quadratic polynomial approach sounds rather like "stationary phase" theory. For exponential sums this is supposed to register with the work of Van der Corput.

Is there a reason to think much can be done for general g?

Surely Bruce Berndt has documented these? The only question would be where.

I've edited the post

Actually Swinnerton-Dyer doesn't stand on ceremony, despite the baronetcy. But the first name is unclear.

So is everyone, however My personal view is based on my own research (actually I started with the Grothendieck–Katz p-curvature conjecture en.wikipedia.org/wiki/Grothendieck-Katz_p-curvature_conjectu‌​re) which is that the local-global view encourages new research. You are reading mathematics written about 50 years ago, and asking a historians' question. Why not find your own problem? This worked for me, and Zariski-dense subgroups. Try works by Emmanuel Kowalski for a fresher approach.

I disagree with at least one of the above comments.

Compute the case where the numbers are in geometric progression?