 Solution

Write $d=|\Gamma |$ and identify each character with a residue $r_j\in \mathbb{Z}/N\mathbb{Z}$. Partition the $d$-dimensional torus into $Q^d$ cubes of side $1/Q$, where $Q$ is comparable to $N^{1/d}$. Applying the pigeonhole principle to the pointsgives a nonzero residue $q$ satisfyingConsequently $|e^{2\pi imq{r}_j/N}-1|\le \rho$ whenever $|m|\le \rho Q/(4\pi )$. After allowing for integer parts and the small values of $Q$, this produces the centered arithmetic progressionof length at least $\frac{1}{8}\rho N^{1/d}$.

Solved by gpt-5.6-sol high.