Let $f$ be an entire function and $L$ a lattice in $\Bbb C$. If, for any $w \in L$, there is a polynomial $P_w$ such that $f(z + w) = f(z) + P_w(z)$ then $f$ is a polynomial.
If I can show that the degree of $\{p_w(z)| w \in L\}$ has a bound say $k \in \Bbb N$ then for any $z \in \Bbb C$ $\exists z_0 \in L_0$ where $L_0$ is the fundamental lattice such that $f(z)=f(z_0)+p_w(z)$ then $|f(z)| < c|z|^k$ for large $z \in \Bbb C$. Then by First Liouville Theorem $f$ would be a constant.
Now the problem is how to show "the degree of $\{p_w(z)| w \in L\}$ has a bound".