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We establish an exponential error term for the renewal theorem in the contextof products of random matrices, which is surprising compared with classicalabelian cases. A key tool is the Fourier decay of the Furstenberg measures onthe projective spaces, which is a higher dimensional generalization of a recentwork of Bourgain-Dyatlov.

Fourier decay, Renewal theorem and Spectral gaps for random walks on split semisimple Lie groups