Monic integer Chebyshev problem

We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let Mn(Z) denote the monic polynomials of degree n with integer coefficients. A monic integer Chebyshev polynomial Mn ∈ Mn(Z) satisfies ‖Mn‖E = inf Pn∈Mn(Z) ‖Pn‖E . and the monic integer Chebyshev constant is then defined by tM (E) := lim n→∞ ‖Mn‖ E . This is the obvious analogue of the more usual integer Chebyshev constant that has been much studied. We compute tM (E) for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases. Conjecture. Suppose [a2/b2, a1/b1] is an interval whose endpoints are consecutive Farey fractions. This is characterized by a1b2 − a2b1 = 1. Then tM [a2/b2, a1/b1] = max(1/b1, 1/b2). 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.