Year: 2026
Type: article
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).
Source: Research in Number Theory
Authors: Faustin Adiceam, Victor Shirandami
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,590
Award: G654398119
OpenAlex ID: W7123935906