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If$X$is a closed subset of the real line, denote by$G$_{$X$}the supremum of the size of the gap in the Fourier spectrum of a measure, taken over all non-trivial finite complex measures supported on$X$. In this paper we attempt to find$G$_{$X$}in terms of$X$.

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We characterize quasiconformal mappings as those homeomorphisms between two metric measure spaces of locally bounded geometry that preserve a class of quasiminimizers. We also consider quasiconformal mappings and densities in metric spaces and give a characterization of quasiconformal mappings in terms of the uniform density property introduced by Gehring and Kelly.

In this note we show that if a projective manifold admits a Khler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.