  In this paper, we first study the strong Birkhoff Ergodic Theorem for
subharmonic functions with the Brjuno-R\"ussmann shift on the Torus. Then, we
apply it to prove the large deviation theorems for the finite scale Dirichlet
determinants of quasi-periodic analytic Jacobi operators with this frequency.
It shows that the Brjuno-R\"ussmann function, which reflects the irrationality
of the frequency, plays the key role in these theorems via the smallest
deviation. At last, as an application, we obtain a distribution of the
eigenvalues of the Jacobi operators with Dirichlet boundary conditions, which
also depends on the smallest deviation, essentially on the irrationality of the
frequency.
