I am trying to determine all entire functions $f$ on the complex plane which satisfy $f(\exp(\sqrt{2}\pi in))=1$ for all $n \in \mathbb{N}$. I observed that the inner function $\exp(\sqrt{2}\pi in)$ maps $n$ to the boundary of the unit disk. But this does not have to be the case for a random $z\in \mathbb{C}$, so I am not sure if this observation helps.
To break down the problem I distinguished two cases: If $f$ is bounded, then $f$ is obviously constant and $f\equiv 1$ by Liouville's theorem. If $f$ is not bounded, it is either a nonconstant polynomial or a transcendental entire function. I feel like the identity principle will not be useful here, because the sequence $(\exp(\sqrt{2}\pi in))_n$ does not converge. I also thought about using Casorati Weierstraß' theorem, but could not figure out in which way.
I am looking for just a tiny hint on which theorem could be useful here or which detail to look at, not a complete solution. Many thanks in advance.
The set $A = \{ \exp(\sqrt{2}\pi in) \mid n \in \Bbb N \}$ is an infinite subset of the (compact) unit circle, and therefore has an accumulation point. Therefore $\{ z \in \Bbb C \mid f(z) = 1 \} \supset A$ has an accumulation point in $\Bbb C$, and the identity theorem can be applied.