The distribution of spacings between small powers of a primitive root

We examine the distribution of spacings between the first N powers of a primitive root modulo a large prime p, where N ≪ p^1-ε. We aim to show that the distribution of these spacings follows the random (or Poisson) model. We show that the m-level correlation functions are Poissonian if JV ≫ p^1-1/2m+ε. For the pair correlation function (m = 2) we can do better: the exponent 3/4 may be replaced by 5/7 unconditionally, and by 2/3 on GRH. Moreover, we show unconditionally that the exponent 2/3 is valid for almost all choices of primitive roots.