Probabilistic Effectivity in the Subspace Theorem

TL;DR

This paper develops a probabilistic framework to analyze the effectiveness of the Subspace Theorem in Diophantine approximation, providing new estimates and density results for algebraic forms and solutions.

Contribution

It introduces probabilistic estimates for the Subspace Theorem, extending applicability to more general approximation functions and refining results for Roth's Theorem.

Findings

Established probabilistic estimates for algebraic forms with bounded heights and degrees.

Derived density results related to the Waldschmidt conjecture.

Extended the theorem's applicability to broader classes of approximation functions.

Abstract

The Subspace Theorem due to Schmidt (1972) is a broad generalisation of Roth's Theorem in Diophantine Approximation (1955) which, in the same way as the latter, suffers a notorious lack of effectivity. This problem is tackled from a probabilistic standpoint by determining the proportion of algebraic linear forms of bounded heights and degrees for which there exists a solution to the Subspace Inequality lying in a subspace of large height. The estimates are established for a class of height functions emerging from an analytic parametrisation of the projective space. They are pertinent in the regime where the heights of the algebraic quantities are larger than those of the rational solutions to the inequality under consideration, and are valid for approximation functions more general than the power functions intervening in the original Subspace Theorem. These estimates are further refined in the case of Roth's Theorem so as to yield a Khintchin-type density version of the so-called Waldschmidt conjecture (which is known to fail pointwise). This answers a question raised by Beresnevich, Bernik and Dodson (2009).