Diophantine properties of Lehmer's continued cotangent expansions
D. H. Lehmer used the function \[ F(x,y)= \frac{xy+1}{y-x}= \cot (\text{ arccot(x)}-\text{ arccot}(y)) \] to devise an algorithm based on the successive iterations \[ F(n_0, F(n_1,F(n_2,\dots)))=\cot \left( \sum_{j \geq 0} (-1)^j \text{ arccot}(n_j)\right). \] The algorithm goes this way: Let \(X=X_0\) be a real number and let \(n_0=[X_0]\). Then recursively define \[ X_{s-1}= \frac{n_sX_s+1}{X_s-n_s} \quad \text{with} \quad n_s=[X_s]. \] In the paper under review, T. Rivoal relates the rational approximations of \(X_0\) coming from this algorithm to the usual convergents of \(X_0\). In particular, he gives an almost optimal bound for the growth of the sequence \((n_s)_{s \geq 0}\) when \(X_0\) is an algebraic number. Many other interesting properties are also exhibited.
Lehmer's continued cotangent developments
scientific article; zbMATH DE number 5133246