I wrote a patch to fix the issues from bugs #20619 and #21348 for GTK
users. When the functions to display a tooltip or menu are called, Emacs
scales coordinates using a factor from GTK. In my testing, non-GTK
tooltips and menus weren't broken, so the problem is specific to GTK and
the patch has no effect on non-GTK builds. Michael Droettboom, will you
apply this patch and verify that the menus are now placed correctly on
your system?

There's something else entirely going on with the scroll bars in bug
#21469, this patch doesn't address that at all. I had never noticed that
hidpi bug because I dont use scroll bars, but I can confirm that turning
on scroll bars causes strange behavior. It might be possible that a
similar scaling strategy for scroll bar placement could provide a fix,
so I CC'd that bug. I will investigate that more as time allows.

The final hidpi bug I looked at, #18429, I am unable to
reproduce. Perhaps it is not applicable to my platform - I'm on Ubuntu
Trusty, while the reporter is on Utopic. Anders Kaseorg, can you still
reproduce the bug?

Finally, there's the open question of why the coordinates these
functions are getting are doubled in the first place. Given my limited
familiarity with Emacs internals, I have not made any progress on that
question. Perhaps there are few enough places where these
sometimes-inflated coordinates are passed into GTK that we can just
scale them everywhere and call it good enough. Or perhaps there's a more
robust solution somewhere else - if anybody can help explain this to me,
I would be appreciative.