Large Sieve Inequalities via Subharmonic Methods and the Mahler Measure of the Fekete Polynomials

Abstract We investigate large sieve inequalities such as $$\frac{1}{m}\underset{j=1}{\overset{m}{\mathop{\sum }}}\,\,\psi \left( \log \,|P({{e}^{i\tau j}})| \right)\,\le \,\frac{C}{2\pi }\,\int_{0}^{2\pi }{\psi }\,\left( \log [e\,|P({{e}^{i\tau }})|] \right)\,d\tau ,$$ where $\psi$ is convex and increasing, $P$ is a polynomial or an exponential of a potential, and the constant $C$ depends on the degree of $P$ , and the distribution of the points $0\,\le \,{{\tau }_{1}}\,<\,{{\tau }_{2}}\,<\,\cdots \,<\,{{\tau }_{m}}\,\le \,2\pi$ . The method allows greater generality and is in some ways simpler than earlier ones. We apply our results to estimate the Mahler measure of Fekete polynomials.