Exponential sums method for frequencies of most significant bit $r$-patterns in linear recurrent sequences over $\mathbb{Z}_{2^n}$

By means of exponential sums method we investigate distributions of $r$-patterns in the most significant bit of linear recurrent sequences over $\mathbb{Z}_{2^n}$ such that their characteristic polynomials reduced to mod $2$ are irreducible over $GF(2)$.