Intrinsic Diophantine Approximation on General Polynomial Surfaces

TL;DR

This paper investigates the measure and dimension of points on polynomial surfaces that can be approximated by rationals, extending previous results from curves to higher-dimensional algebraic manifolds.

Contribution

It provides a comprehensive analysis of intrinsic Diophantine approximation on polynomial surfaces, generalizing prior curve-based results to more complex algebraic manifolds.

Findings

Determined Hausdorff measure and dimension for approximable points on polynomial surfaces.

Extended Diophantine approximation results from curves to higher-dimensional algebraic manifolds.

Achieved complete characterizations for algebraically 'nice' manifolds.

Abstract

We study the Hausdorff measure and dimension of the set of intrinsically simultaneously $\psi$-approximable points on a curve, surface, etc., given as a graph of integer valued polynomials. We obtain complete answers to these questions for algebraically "nice" manifolds. This generalizes earlier work done in the case of curves.