# The Fuglede conjecture holds in the finite vector space $$\mathbb{Z}_p^2$$

### The Fuglede conjecture holds in the finite vector space $$\mathbb{Z}_p^2$$

We will see that the Fuglede Conjecture holds in $\mathbb{Z}_p^2$, proved by Iosevich/Mayeli/Pakianathan. That is the subsets E of the finite vector space $\mathbb{Z}_p^2$ tiles the space if and only if every function from E to the complex numbers is a linear combination of orthogonal exponential functions. Key ideas of the proof use Fourier transforms of functions from this space, direction sets, and some Galois theory.