  We prove $\times a$ $\times b$ measure rigidity for multiplicatively
independent pairs when $a\in\mathbb{N}$ and $b>1$ is a ``specified'' real
number (the $b$-expansion of $1$ has a tail or bounded runs of $0$'s) under a
positive entropy condition. This is done by proving a mean decay of the Fourier
series of the point masses average along $\times b$ orbits. We also prove a
quantitative version of this decay under stronger conditions on the $\times a$
invariant measure. The quantitative version together with the $\times b$
invariance of the limit measure is a step toward a general Host-type pointwise
equidistribution theorem in which the equidistribution is for Parry measure
instead of Lebesgue. We show that finite memory length measures on the
$a$-shift meet the mentioned conditions for mean convergence. Our main proof
relies on techniques of Hochman.
