Distribution of the zeros of polynomials near the unit circle

We estimate the number of zeros of a polynomial in C[z] within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{\'e}lyi, and Littmann~{BE1} in 2008. Furthermore, by combining this result with Euclidean geometry, we derive an upper bound on the number of zeros of such a polynomial within a region resembling a gear wheel. Additionally, we obtain a sharp upper bound on the annular discrepancy of such zeros near the unit circle. Our approach builds upon a modified version of the method described in {BE1}, combined with the refined version of the best-known upper bound for angular discrepancy of zeros of polynomials.