On the determinant of an integral lattice generated by rational approximants of the Euler constant

We investigate rational approximants of the Euler constant constructed using a certain system of jointly orthogonal polynomials and “averaging” such approximants of mediant type. The properties of such approximants are related to the properties of an integral lattice in R3 constructed from recurrently generated sequences. We also obtain estimates on the metric properties of a reduced basis, which imply that the γ-forms with coefficients constructed from basis vectors of the lattice tend to zero. The question of whether the Euler constant is irrational is reduced to a property of the bases of lattices.