Consider the function $f\colon [0,1]\to\mathbb{R}$ defined by $$f(x)=\frac{\sin(9\pi x)\sin(100\pi x)}{\sin(\pi x)\sin(90\pi x)}.$$ I want to bound this function for certain values of $x$ but I don't know how to make a more precise statement. I'm also only interested in $|f(a/10^m)|$ where $m$ is a fixed positive integers and $a$ is an integer such that $0\lt a \lt 10^m$.
What I think is true is that $|f(a/10^m)|\le 10$ for almost all values $a$. More precisely, $|f(x)|$ is big only when $\sin(90\pi x)=0$, therefore $|f(a/10^m)|$ is big when $90\pi a/10^m$ is a multiple of $\pi$, i.e., the problem will be when $a\approx n\cdot \frac{10^{m-1}}{9}$ ($n=1,\ldots, 90$) so for instance if $m=3$, $|f(a/10^3)|$ is big only when $a$ is close to multiples of $11$. Now my question is how close does $a$ has to be to $n\cdot \frac{10^{m-1}}{9}$ in order for $|f(a/10^m)|$ to be greater than $10$.
Is it possible to make an statement as follows? For every $n=1,\ldots, 90$ there exists some $\varepsilon_n$ such that if $|a-n\cdot \frac{10^{m-1}}{9}|\gt \varepsilon_n$, then $|f(a/10^m)|\le 10$. And if so, how can I prove this?