Bounds for the smallest limit point of the areal Mahler measure of totally positive algebraic integers

Pritsker's areal analog ‖P‖₀ of the Mahler measure replaces the integral of log|P| over the unit circle by its integral over the unit disc with respect to normalized area. For an algebraic number α of degree n with minimal polynomial P in Z[x], taken primitive with positive leading coefficient, let N₀(α) = ‖P‖₀^1/n. Flammang proved that N₀(α) ≥ 1.380047 for every totally positive algebraic integer α outside an explicit finite set. We prove that the smallest limit point of the values of N₀ on totally positive algebraic integers and their essential minimum lie between 1.3830134 and 1.3839579. The lower bound improves on Flammang's constant: it holds for all but finitely many totally positive algebraic numbers, not only integers, although the exceptions are not determined. Infinitely many totally positive algebraic integers satisfy N₀(α) < 1.3839579. The lower bound comes from an auxiliary function with a logarithmic energy term, and the upper bound from an explicit measure that, by a criterion and a theorem of Smith, is a limit of distributions of conjugates of totally positive algebraic integers. The inequalities behind both bounds are verified in ball arithmetic, and both bounds are also proved in Lean, the upper one assuming Smith's theorem.

The certificates, the programs that check them and the Lean proofs are archived in the supplement record doi:10.5281/zenodo.23715998 and in the repository github.com/tejasvi/areal-mahler-measure (tag v10).