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

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