Bernoulli convolution is absolutely continuous for l=1-10^(-50) and 3/4>=p>=1/4

You need to know: Basic probability theory, notation $\text{Pr}$ for probability, random variable, distribution of a random variable, independent identically distributed (i.i.d.) random variables, polynomial, degree of a polynomial, monic polynomial. You also need Lebesgue measure and Hausdorff dimension to understand the context.

Background: Let $\lambda,p\in(0,1)$ be real numbers, and let $\xi_0, \xi_1, \xi_2, \dots$ be a sequence of i.i.d. random variables with $\text{Pr}(\xi_n=1)=p$ and $\text{Pr}(\xi_n=-1)=1-p$ for all n. By Bernoulli convolution we mean the random variable $X_{\lambda,p}=\sum\limits_{n=0}^\infty \xi_n \lambda^n$ . $X_{\lambda,p}$ is called absolutely continuous if there exists function $f:{\mathbb R}\to{\mathbb R}$ such that $\text{Pr}(a\leq X_{\lambda,p}\leq b)=\int_a^b f(x)dx$ whenever $a\leq b$ . The Mahler measure of any polynomial $P(x)=a\prod\limits_{i=1}^d(x-z_i)$ is $M(P)=a\prod\limits_{j:|z_j|>1}|z_j|$ . A real number $r$ is called algebraic if $P(r)=0$ for some polynomial P with rational coefficients. The (unique) monic polynomial $P_r$ of smallest degree with this property is called the minimal polynomial of r. The Mahler measure of r is $M_r=M(P_r)$ .

The Theorem: On 31st January 2016, Péter Varjú submitted to arxiv a paper in which he proved that for every $\epsilon>0$ and $p\in(0,1)$ , there is a constant $c=c(\epsilon,p)>0$ such that for any algebraic number $\lambda$ satisfying $1>\lambda>1 - c\min(\log M_\lambda, (\log M_\lambda)^{-1-\epsilon})$ , the Bernoulli convolution $X_{\lambda,p}$ is absolutely continuous.

Short context: Bernoulli convolutions has been introduced by Jessen and Wintner in 1935, and studied by Erdős and many others since that. The main research question is for which values of $\lambda$ and $p$ the Bernoulli convolution is absolutely continuous. In 1939, Erdős noticed that $X_{\lambda,1/2}$ is not absolutely continuous for $\lambda<1/2$ , and also for $\lambda \in E$ for some non-empty set $E \subset [1/2,1)$ . However, Solomyak proved in 1995 that set E has Lebesgue measure $0$ . Moreover, Shmerkin, building on this theorem, proved in 2014 that E has Hausdorff dimension $0$ . Despite on this, it was known very few explicit examples of $\lambda$ -s for which $X_{\lambda,p}$ is absolutely continuous, and none such examples was known for $p\neq 1/2$ . The Theorem (with explicit $c=c(\epsilon,p)$ ) provides a lot of such examples. For example, it implies that $X_{\lambda,p}$ is absolutely continuous for $\lambda=1-10^{-50}$ and $\frac{1}{4}\leq p \leq \frac{3}{4}$ .