Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties

TL;DR

This paper introduces an invariant measuring rational point approximation on algebraic varieties, linking it to local positivity and generalizing Roth's theorem beyond projective spaces.

Contribution

It establishes a new invariant related to Diophantine approximation and connects it to Seshadri constants, extending Roth's theorem to arbitrary varieties.

Findings

Invariant $eta_x(L)$ quantifies approximation quality.

Relation between $eta_x(L)$ and Seshadri constant $ ext{epsilon}_x(L)$.

Generalization of Roth's theorem to all projective varieties.

Abstract

In this paper, we associate an invariant $\alpha_{x}(L)$ to an algebraic point $x$ on an algebraic variety $X$ with an ample line bundle $L$. The invariant $\alpha$ measures how well $x$ can be approximated by rational points on $X$, with respect to the height function associated to $L$. We show that this invariant is closely related to the Seshadri constant $\epsilon_{x}(L)$ measuring local positivity of $L$ at $x$, and in particular that Roth's theorem on $\mathbf{P}^1$ generalizes as an inequality between these two invariants valid for arbitrary projective varieties.