We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
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.
