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Large sieve inequalities via subharmonic methods and the Mahler measure of the Fekete polynomials. (English) Zbl 1166.41004

Summary: We investigate large sieve inequalities such as
\[ \frac{1}{m} \sum_{j=1}^m \psi(\log|P(e^{i \tau_j})|) \leq \frac{C}{2 \pi} \int_0^{2 \pi} \psi(\log[e|P(e^{i \tau})|])\,d \tau, \]
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.

MSC:

11C08 Polynomials in number theory
11N35 Sieves
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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