Monic integer Chebyshev problem

We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let ${\mathcal{M}}_n({\mathbb{Z} })$ denote the monic polynomials of degree $n$ with integer coefficients. A monic integer Chebyshev polynomial $M_n \in {\mathcal{M}}_n({\mathbb{Z} })$satisfies

and the monic integer Chebyshev constant is then defined by

This is the obvious analogue of the more usual integer Chebyshev constant that has been much studied.

We compute $t_M(E)$ for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases.

Conjecture. Suppose $[{a_2}/{b_2},{a_1}/{b_1}]$ is an interval whose endpoints are consecutive Farey fractions. This is characterized by $a_1b_2-a_2b_1=1.$ Then

This should be contrasted with the nonmonic integer Chebyshev constant case, where the only intervals for which the constant is exactly computed are intervals of length 4 or greater.