Consider the product $P_n(x) = \sin(x-a)\sin(2x-a)\cdots\sin(2^n x - a)$, where $a \in [0,\pi/2]$.What is the asymptotic of $\max |P_n(x)|$. More precisely, is there a positive constant$A = A(a)$, such that $A^{n+1} \leq \max |P_n(x)| \leq A^n$?
Obviously, $A(\pi/2) = 1$ and it is known that $A(0) = \sqrt{3}/2$.
I was not able to find any information related to this question for other values of $a$.