  Motivated by the Rudnick-Sarnak theorem we study limiting distribution of
smoothed local correlations of the form $$ \sum_{j_1, j_2, \ldots, j_n}
f(N\*(\theta_{j_2}-\theta_{j_1}), N\*(\theta_{j_3}-\theta_{j_1}), \ldots,
N\*(\theta_{j_n}-\theta_{j_1}))$$ for the Circular United Ensemble of random
matrices for sufficiently smooth test functions.
