Tarski proved that equalities and inequalities in can be decided over $\mathbb{R}[x].$ Richardson proved that adding composition with the sine and exponential functions caused the problem to become undecidable. What are the best results sharpening this gap?
Laczkovich (improving Wang) was able to show undecidability for the ring generated by the integers, $x$, $\sin x^n,$ and $\sin(x\sin x^n)$ which is the best result I know of in this direction. I don't know of any results on the decidable side.
In particular, is it known whether inequalities over the ring $\mathbb{Z}[x,\sin x]$ are decidable?