Decrease of Fourier coefficients of Furstenberg measures and renewal theory

Let mu be a Borel probability measure on SL(2,R) with a finite 
exponential moment, such that the support of mu generates a Zariski 
dense subgroup in SL(2,R). We can define a unique probability measure on 
the circle, which is called the mu stationary measure or Furstenberg 
measure. We will prove, using Bourgain’s discretized sum-product 
estimate, that the Fourier coefficients of this measure go to zero with 
a polynomial speed. Starting from this result, we can obtain a spectral 
gap of the transfer operator, whose properties enable us to get an 
exponential error term in the renewal theorem in the context of random 
products of matrices.