2 Some explicit badly approximable pairs

I consider the Diophantine approximation problem of sup-norm simultaneous rational approximation with common denominator of a pair of irrational numbers, and compute explicitly some pairs with large approximation constant. One of these pairs is the most badly approximable pair yet computed. The theory of approximation of a single irrational number by rationals is well known, and for our purposes the relevant facts may be summarized as follows. We measure the goodness of approximation of the rational number p/q to α by c(α, p, q) ≡ q|qα − p|. For each irrational α (without loss of generality, we may assume 0 < α < 1) we know by Dirichlet's thereom that there are infinitely many rationals p/q such that |α−p/q|<1/q 2 , or c(α, p, q)<1. It is therefore of interest to ask how small one may make γ in c(α, p, q)<γ before this property fails to hold. The approximation constant of α is thus defined as c(α) ≡ lim inf q→∞ c(α, p, q). Here, of course, for each q we choose the p whch minimizes c(α, p, q). Numbers α with a large c(α) are hard to approximate by rationals. The one-dimensional Diophantine approximation constant , defined as c 1 = sup α∈R c(α), has the value 1/ √ 5, attained at α = (√ 5−1)/2. Otherwise expressed, this means that c 1 is the unique number such that for each ε >0, the inequality c(α, p, q) < c 1 +ε has infinitely many rational solutions p/q for all α, whereas there is at least one α such that c(α, p, q) < c 1 −ε has only finitely many rational solutions.