  This note contains a new proof of Host's equidistribution theorem for
multiplicatively independent endomorphisms of $\mathbb{R}/\mathbb{Z}$. The
method is a simplified version of our upcoming work on equidistribution under
toral automorphisms and is related to the argument in [HochmanShmerkin2015],
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.
