The exact overlaps conjecture for self-similar measures

Let $\Phi=\{\varphi_j(x)=\lambda_jx+t_j\}_{j=0}^m$ be a self-similar iterated function system on the real line, let $p=(p_j)_{j=0}^m$ be a probability vector, and let $\mu$ be its associated self-similar measure. Write

$$H(p)=-\sum_{j=0}^m p_j\log p_j,\qquad \chi=-\sum_{j=0}^m p_j\log|\lambda_j|,$$

where the logarithms have base $2$, and set

$$\beta=\min\{1,H(p)/\chi\}.$$

Denote $\{0,\ldots,m\}$ by $\Lambda$. The IFS has exact overlaps if there exist $n\geq 1$ and distinct words $w_1,w_2\in\Lambda^n$ such that $\varphi_{w_1}=\varphi_{w_2}$. Exact overlaps conjecture. If $\dim\mu<\beta$, then $\Phi$ has exact overlaps.

The conjecture asserts that exact overlaps are the only mechanism for dimension drop below the entropy--Lyapunov bound. A version for self-similar sets was attributed in the source to Simon; the conjecture is the subject of the paper, which proves it for systems with algebraic contractions.