A converse to linear independence criteria, valid almost everywhere

TL;DR

This paper establishes a weighted version of the Khintchine-Groshev Theorem and uses it to demonstrate the optimality of certain linear independence criteria over rationals.

Contribution

It introduces a weighted analogue of a classical theorem and applies it to validate the best possible linear independence criteria.

Findings

Weighted Khintchine-Groshev Theorem proved

Optimality of linear independence criteria established

Applications to rational number linear independence

Abstract

We prove a weighted analogue of the Khintchine-Groshev Theorem, where the distance to the nearest integer is replaced by the absolute value. This is subsequently applied to proving the optimality of several linear independence criteria over the field of rational numbers.