On two problems of Erdos and Hechler: New methods in singular madness

For an infinite cardinal $\mu$, $\operatorname{MAD}(\mu)$denotes the set of all cardinalities of nontrivial maximal almost disjoint families over $\mu$.

Erdos and Hechler proved in 1973 the consistency of $\mu\in \operatorname{MAD}(\mu)$for a singular cardinal $\mu$ and asked if it was ever possible for a singular $\mu$ that $\mu\notin \operatorname{MAD}(\mu)$, and also whether $2^{\operatorname{cf}\mu} <\mu \Longrightarrow \mu\in \operatorname{MAD}(\mu)$ for every singular cardinal $\mu$.

We introduce a new method for controlling $\operatorname{MAD} (\mu)$ for a singular $\mu$ and, among other new results about the structure of $\operatorname{MAD}(\mu)$ for singular $\mu$, settle both problems affirmatively.