Free entropy dimension and regularity of non-commutative polynomials

We show that the spectral measure of any non-constant non-commutative polynomial evaluated at 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 of any non-constant non-commutative polynomial function is not singular, and measures of intervals surrounding any point may not decay slower than polynomially as a function of the interval's length. (C) 2016 Elsevier Inc. All rights reserved.