A **Rajchman measure** on the unit circle $\mathbb{T}$ is a Borel probability measure $\mu$ with $\lim\_{n\to\infty}\hat{\mu}(n)=0$. Where $\hat{\mu}(n)=\mu(z^n)$ for $n\in\mathbb{Z}$ are Fourier coefficients of $\mu$.


Suppose $(X,\mathcal{B},T,\mu)$ is a measure-theoretic dynamical system consisting of a measure space $(X,\mathcal{B},\mu)$ and an invertible measure preserving map $T:X\to X$. A dynamical system $(X,\mathcal{B},T,\mu)$ is called **strong mixing** if
$$\lim\_{n\to\infty} \mu(T^{-n}A\cap B)=\mu(A)\mu(B)$$ for all $A,B\in\mathcal{B}$.


Denote the space $\{f\in L^2(X,\mu)|\int\_X f\,d\mu=0\}$ by $L^2\_0(X,\mu)$. The dynamical system $(X,\mathcal{B},T,\mu)$ is strong mixing iff
$$\lim\_{n\to\infty}\int\_X f(T^nx)\overline{f(x)}\,d\mu(x)=0$$ for every unit $f\in L^2\_0(X,\mu)$.


For every unit $f\in L^2\_0(X,\mu)$, one can define a Borel probability measure $\mu\_f$ on $\mathbb{T}$
by $$\widehat{\mu\_f}(n)=\int\_X f(T^nx)\overline{f(x)}\,d\mu(x).$$ So a dynamical system $(X,\mathcal{B},T,\mu)$ is strong mixing iff $\mu\_f$ is a Rajchman measure for every unit $f\in L^2\_0(X,\mu)$.


Through the above observation, given a strong mixing dynamical system $(X,\mathcal{B},T,\mu)$, for instance, Bernoulli shift $(\{0,1\}^\mathbb{Z},\mathcal{B},S,\mu\_p)$ where $\mu\_p$ is the product measure of the measure on $\{0,1\}$ giving $\{0\}$ measure $p$ and $\{1\}$ measure $1-p$, each $f\in L^2\_0(X,\mu)$ gives rise to a Rajchman measure on $\mathbb{T}$.


**Question**: For which strong mixing dynamical system $(X,\mathcal{B},T,\mu)$, one can find unit $f\in L^2\_0(X,\mu)$ such that $\mu\_f$ is a Rajchman measure singular to the Lebesgue measure? 



Rajchman measures via strong mixing systems
I think you're exactly asking for the strong mixing dynamical systems with singular spectral type. One comment: these must have entropy 0 as any positive entropy measure has Lebesgue spectrum.


