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

We investigate large sieve inequalities such as $\frac{1}{m}\sum_{j=1}^{m}\psi ( \log | P( e^{i\tau _{j}}) | ) \leq \frac{C}{2\pi }\int_{0}^{2\pi }\psi \left( \log [ e| P( e^{i\tau }) | ] \right) \,d\tau ,$ 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 $0\leq \tau _{1}<\tau _{2}<\dots<\tau _{m}\leq 2\pi$. 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.