Borel's conjecture on binary string complexity of algebraic numbers

Let $r\in(0,1)$ be an irrational algebraic real, and let $p_r(m)$ denote the number of distinct binary words of length $m$ occurring in the binary expansion of $r$. Borel's conjecture. The binary string complexity of $r$ should satisfy

$$p_r(m)=2^m.$$

This conjecture asserts that every possible binary word of each length occurs in the expansion of an irrational algebraic number. It was proven in 2007 by B. Adamczewski and Y. Bugeaud.