Simultaneous Diophantine approximation: sums of squares and homogeneous polynomials

TL;DR

This paper investigates uniform simultaneous Diophantine approximation on the graph and hypersurface of homogeneous polynomials with rational coefficients, focusing on sums of squares and general homogeneous forms.

Contribution

It extends Diophantine approximation results to the setting of homogeneous polynomials, including sums of squares, providing new approximation bounds.

Findings

Results for approximation on the sum of squares hypersurface.

Generalization to arbitrary homogeneous polynomials.

Specific bounds for approximation quality.

Abstract

Let $f$ be a homogeneous polynomial with rational coefficients in $d$ variables. We prove several results concerning uniform simultaneous approximation to points on the graph of $f$, as well as on the hypersurface $\{f(x_1,\dots,x_d) = 1\}$. The results are first stated for the case $f(x_1,\dots,x_d) = x_1^2+\dots+x_d^2,$ which is of particular interest.