The paper proves, in self-contained form, that any non-atomic self-similar measure whose Fourier transform vanishes at infinity is pointwise absolutely normal.
The coincidence of R\'{e}nyi-Parry measures for $\beta$-transformation
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abstract
We present a complete characterization of two different non-integers with the same R\'{e}nyi-Parry measure. We prove that for two non-integers $\beta_1 ,\beta_2 >1$, the R\'{e}nyi-Parry measures coincide if and only if $\beta_1$ is the root of equation $x^2-qx-p=0$, where $p,q\in\mathbb{N}$ with $p\leq q$, and $\beta_2 = \beta_1 + 1$, which confirms a conjecture of Bertrand-Mathis in \cite[Section III]{Bertrand-1998}.
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Recent progress on pointwise normality of self-similar measures
The paper proves, in self-contained form, that any non-atomic self-similar measure whose Fourier transform vanishes at infinity is pointwise absolutely normal.