On Newman and Littlewood polynomials with a prescribed number of zeros inside the unit disk

We study \{0, 1\} and \{-1, 1\} polynomials f(z), called Newman and Littlewood polynomials, that have a prescribed number N(f) of zeros in the open unit disk \mathbb{D} = \{z \in \mathbb{C}: \lvert z \rvert < 1\}. For every pair (k, n) \in \mathbb{N}^2, where n \geq 7 and k \in [3, n-3], we prove that it is possible to find a \{0, 1\}-polynomial f(z) of degree \deg {f}=n with non-zero constant term f(0) \ne 0, such that N(f)=k and f(z) \ne 0 on the unit circle \partial \mathbb{D}. On the way to this goal, we answer a question of D. W. Boyd from 1986 on the smallest degree Newman polynomial that satisfies \lvert f(z) \rvert > 2 on the unit circle \partial \mathbb{D}. This polynomial is of degree 38 and we use this special polynomial in our constructions. We also identify (without a proof) all exceptional (k, n) with k \in \{1, 2, 3, n-3, n-2, n-1\}, for which no such \{0, 1\}-polynomial of degree n exists: such pairs are related to regular (real and complex) Pisot numbers. Similar, but less complete results for \{-1, 1\} polynomials are established. We also look at the products of spaced Newman polynomials and consider the rotated large Littlewood polynomials. Lastly, based on our data, we formulate a natural conjecture about the statistical distribution of N(f) in the set of Newman and Littlewood polynomials.