A non trivial example of a spread out distribution This question is motivated by this posting.
In the classic book Meyn, S. and Tweedie, R., Markov Chains and Stochastic Stability, the authors introduce the concept of a spread out distribution:

Definition:
A probability measure $\mu$ on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$ is a spread out distribution if there is $n\in\mathbb{N}$ such that $\mu^{*n}$ has a nonzero absolutely continuous  (w.r.t. Lebesgue measure).

Here $\mu^{*n}$ denotes the convolution of $\mu$ with itself $n$-times. These measures come as examples of random walks where ergodicity properties can be analyze in terms of small sets.
Problem: Any measure $\nu$ with non trivial absolutely continuous part is of course spread out ($n=1$). What I am asking (out of curiosity and not for professional need) is for an example of a purely singular measure $\nu$ (w.r.t Lebesgue measure) that is spread out in the sense above.
My first instinct was look for example,  the 1/3-Cantor measure $\mu_{1/3}$. Yuval Peres, here show me that this measure (and other fractal-like measures) preserve singularity under self-convolutions. Thus, such $\nu$ must be either too easy to obtain for my  myopic eyes, or much more sophisticated.
 A: Given $\lambda \in (0,1)$, consider the Bernoulli convolution $\nu_\lambda$, which is the distribution of the random series $\sum \pm \lambda^n$ (see [1]). For $\lambda<1/2$, this measure is the uniform Cantor-Lebesgue measure on the middle $1-2\lambda$ Cantor set (up to an affine transformation on the real line) so $\nu_\lambda$ is singular for all $\lambda<1/2$.    In Cor. 1.6 page 4070 of [1], it is shown that for a.e. $\lambda>3/8$, The Fourier transform of  $\nu_\lambda$ is in $L^4$, so
the convolution $\nu_\lambda*\nu_\lambda$ is absolutely continuous, with a density in $L^2$. That same corollary implies that for a.e. $\lambda >5/16$, the convolution $\nu_\lambda^{*3}$ is absolutely continuous, etc.
[1] Peres, Yuval, Wilhelm Schlag, and Boris Solomyak. "Sixty years of Bernoulli convolutions." In Fractal geometry and stochastics II, pp. 39-65. Birkhäuser, Basel, 2000.
http://u.math.biu.ac.il/~solomyb/RESEARCH/sixty.pdf
[2]  Peres, Yuval, and Boris Solomyak. "Self-similar measures and intersections of Cantor sets." Transactions of the American Mathematical Society 350, no. 10 (1998): 4065-4087.  https://www.ams.org/journals/tran/1998-350-10/S0002-9947-98-02292-2/S0002-9947-98-02292-2.pdf
