Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 7E / b / ii / Solution

Since $p/q$ is not a root, $q^{d}f(p/q)$ is a nonzero integer, and henceMoreover $0<\theta -p/q<1$ implies $|p/q|<|\theta |+1$. Applying part (i) to $s=\theta$, $t=p/q$, and using $f(\theta )=0$ givesRearrangement provesThis is the core estimate behind the Liouville approximation theorem.

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