A short proof of Host’s equidistribution theorem

This note contains a new proof of Host’s equidistribution theorem for multiplicatively independent endomorphisms of ℝ/ℤ. The method is a simplified version of our recent work on equidistribution under toral automorphisms [2] and is related to the argument in [3], but avoids the use of the scenery flow and of Marstrand’s projection theorem, using instead a direct Fourier argument to establish smoothness of the limit measure.