Regularity of Polynomials in Free Variables

We show that the spectral measure of any non-commutative polynomial of a non-commutative $n$-tuple cannot have atoms if the free entropy dimension of that $n$-tuple is $n$ (see also work of Mai, Speicher, and Weber). Under stronger assumptions on the $n$-tuple, we prove that the spectral measure is not singular, and measures of intervals surrounding any point may not decay slower than polynomially as a function of the interval’s length.