I am considering undertaking some independent research in my summer break studying Gaussian primes in translations of lattices in $\mathbb{Z}[i]$, i.e. sets of the form $ \{ a+sx+tw:s,t \in \mathbb{Z} \}$ where $a,x,w \in \mathbb{Z}[i]$. I would most likely ask myself questions such as when does this set contain infinitely many Gaussian primes (if ever), extending the method of Dirichlet's theorem on primes in arithmetic progressions.
If anybody knows any results relating to Gaussian primes in translations of lattices in $\mathbb{Z}[i]$ please let me know!
