On absolute continuity of the Erdős measure for the golden ratio, Tribonacci numbers, and second-order Markov chains (Q6589449)

For a stationary \(\{0,1\}\)-valued process \((\epsilon_k)\) the associated Erdős measure \(\mu\) is the distribution law of the random variable \(\zeta=\sum_{k=1}^{\infty}\epsilon_k\beta^{-k}\) for \(\beta\in(1,2)\). If the process is a \(0\)-Markov chain (that is, independent and identically distributed) then \(\mu\) is an infinite Bernoulli convolution. The `Erdős problem' in this context is to understand when \(\mu\) is absolutely continuous. The main results here give explicit conditions in terms of transition probabilities for \(1\)-Markov and \(2\)-Markov processes to have absolutely continuous Erdős measures when \(\beta\) is the golden mean and for \(2\)-Markov processes and the Tribonacci number. This completes earlier partial results finding some of the conditions that guarantee absolute continuity.