Sur le diamètre transfini entier d'un intervalle réel. (On the transfinite integer diameter of a real interval) (Q749792)

Let \(X\subset C\) be a compact set; its transfinite integer diameter \(t_ Z(X)\) is defined as a measure for the least deviation from zero of integral polynomials over X. The behaviour of the function \(t_ Z\) over the set of all intervals I is studied. In particular, some bounds for \(t_ Z(I)\) which become sharp for intervals with rational extremities, are given. These bounds, obtained combining the theory of orthogonal polynomials with elementary arguments from the geometry of numbers, explicitly depend on the diameter of I and on the denominators of its extremities.