On the arithmetic case of Vojta's conjecture with truncated counting functions

TL;DR

This paper establishes a Diophantine approximation inequality related to Vojta's conjecture with truncated counting functions, providing bounds that relate rational points, divisors, and algebraic points, with implications for the abc conjecture.

Contribution

It proves a new inequality for rational points on varieties involving truncated counting functions, and links the Lang-Waldschmidt conjecture to special cases of Vojta's conjecture.

Findings

Provides a lower bound for truncated counting functions in terms of proximity to algebraic points.

Establishes a subexponential bound towards the abc conjecture in certain cases.

Shows the Lang-Waldschmidt conjecture implies a case of Vojta's conjecture with truncation.

Abstract

We prove a Diophantine approximation inequality for rational points in varieties of any dimension, in the direction of Vojta's conjecture with truncated counting functions. Our results also provide a bound towards the $abc$ conjecture which in several cases is subexponential. The main theorem gives a lower bound for the truncated counting function relative to a divisor with sufficiently many components, in terms of the proximity to an algebraic point. Furthermore, we show that the Lang-Waldschmidt conjecture implies a special case of Vojta's conjecture with truncation in arbitrary dimension. Our methods are based on the theory of linear forms in logarithms and a geometric construction.