Approximation des moyennes arithmético-géométriques. (Approximation of arithmetic-geometric means) (Q1101501)

Starting from two positive real numbers a,b their arithmetic and geometric means form a new pair \((a_ 1,b_ 1)\) and the iteration of this process gives two sequences \((a_ n)\), \((b_ n)\) such that \(a_{n+1}=(a_ n+b_ n)/2\) and \(b_{n+1}=\sqrt{a_ nb_ n}\). It is known that these sequences converge to a common limit M(a,b), the arithmetico-geometric mean of a and b. Applying the same process to two complex numbers we observe that there is at each step the choice of the square root. The author calls the good choice of \(b_ 1=\sqrt{a_ 0b_ 0}\) that for which \(| a_ 1-b_ 1| \leq | a_ 1+b_ 1|\) and also \(Im(b_ 1/a_ 1)>0\) if there is equality. All sequences obtained from \(a_ 0\) and \(b_ 0\) converge and the limits are \(\neq 0\) if and only if one makes good choices from a certain n on. The author denotes the Mag (moyenne algébrico- géométrique) of a and b by M(a,b) if only good choices are made. The limits distinct from 0 obtained by applying some bad choices at the start are denoted by mag. The author proves: Proposition 1. The mag of two nonzero complex algebraic numbers a and b with \(a\neq \pm b\) are transcendental numbers with type of transcendence at most \(2+\epsilon\) for all \(\epsilon >0\). - Proposition 2. For every integer \(n>14\) there exists a constant c(n) such that for every rational p/q with \(q\geq 2\) \[ | M(1,\sqrt{1-1/n})- p/q| > q^{-c(n)}. \] Proposition 3. If \(a=a_ 0\) and \(b=b_ 0\) are algebraic numbers \(\neq 0\) such that \(a_ 0/b_ 0\) and \(4a_ 1/b_ 1\) are not algebraic units then \(\deg (b_ n)>2^{n-c}\) where c is a constant depending only on a and b.