  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.
