A lower bound for the smallest limit point of the absolute Mahler measure of totally positive algebraic integers

Let ℓ₊ be the smallest limit point of the absolute Mahler measures Ω(α) = M(α)^{1/deg α} of totally positive algebraic integers α. By work of Smyth, ℓ₊ ≤ 1.7274, and the best lower bound was Flammang’s ℓ₊ ≥ 1.722069. We prove that all but finitely many totally positive algebraic numbers α satisfy Ω(α) > 1.7234615, and that ℓ₊ > 1.7234615. We also prove that all but finitely many totally real algebraic numbers β have absolute logarithmic Weil height log Ω(β) > 0.2721673. The proof adds to Smyth’s method of auxiliary functions the logarithmic energy term that Orloski, Talebizadeh Sardari and Smith derived from the integrality of discriminants for the Schur–Siegel–Smyth trace problem. The resulting inequality is verified rigorously by computer, and the lower bounds are also proved in the Lean proof assistant.

Version 4 revises the text of the paper, with a new closing section on open problems and an appendix for the search and the numerical observations. The theorems and their proofs are those of version 3.

The certificate, the checking programs, their tests, the search programs and the Lean files are archived at doi:10.5281/zenodo.23715959 and available at https://github.com/tejasvi/totally-positive-mahler-measure/tree/v4 (commit d03717788979).