We study eigenfunction localization for higher dimensional cat maps
(linear symplectic maps in Sp(2g, Z)), a popular model of quantum
chaos. Under natural assumptions, we show that there is a density one
sequence of integers N so that as N tends to infinity along this
sequence, all eigenfunctions of the quantized map at the inverse
Planck constant N are uniformly distributed. Compared to the the two
dimensional case, higher dimensions requires a new techniques,
including additive combinatorics and properties of tensor product
structures for the cat map. "One good prime is all you need."
