# Expectations, Concave Transforms, Chow weights, and Roth's theorem for varieties

## TL;DR

This paper connects the complexity of rational points on projective varieties with Chow forms and Okounkov bodies, providing new insights into Diophantine approximation and establishing Roth-type theorems using these geometric tools.

## Contribution

It introduces a novel interpretation of rational point complexity via Chow weights and Okounkov bodies, and applies this to prove Roth-type theorems in Diophantine approximation.

## Key findings

- Asymptotic volume and Seshadri constants relate to Chow weights and Okounkov bodies.
- Established Roth-type theorems using normalized Chow weights.
- Connected geometric invariants with arithmetic properties of rational points.

## Abstract

We explain how complexity of rational points on projective varieties can be interpreted via the theories of Chow forms and Okounkov bodies. Precisely, we study discrete measures on filtered linear series and build on work of Boucksom and Chen, Boucksom-et-al and Ferretti. For example, we show how asymptotic volume and Seshadri constants are related to the theories of Chow weights and Okounkov bodies for very ample linear series. We also consider arithmetic applications within the context of Diophantine approximation. In this direction, we establish Roth-type theorems which can be expressed in terms of normalized Chow weights and measures of local positivity.