Extreme values and logarithm laws for toral translations

We study extreme values of normalized $r$-th nearest neighbour distance functions in inhomogeneous Diophantine approximation, proving Weibull and Fréchet limit laws when the shift satisfies a natural Diophantine condition. Using these extreme value laws we prove corresponding logarithm laws for these functions. Our approach is based on homogeneous dynamics. The key input is to adapt the Poisson approximation strategy of Björklund and Gorodnik (arXiv:2201.05116) combined with an effective multiple equidistribution result, which we deduce using results of Strömbergsson (arXiv:1309.6103) and of Björklund and Gorodnik (arXiv:2105.05468).