Moment transference principles and multiplicative diophantine approximation on hypersurfaces

TL;DR

This paper establishes the generic multiplicative approximation rates on hypersurfaces, analyzing four regimes based on convergence/divergence and curvature, through a framework linking Lebesgue and other measures using Fourier analysis.

Contribution

It introduces a general moment transference framework that connects Lebesgue measure data to other measures, applied to multiplicative Diophantine approximation on hypersurfaces.

Findings

Determined approximation rates for all four regimes on hypersurfaces.

Developed a Fourier-based geometric and arithmetic framework.

Unified treatment of convergence and divergence cases.

Abstract

We determine the generic multiplicative approximation rate on a hypersurface. There are four regimes, according to convergence or divergence and curved or flat, and we address all of them. Using geometry and arithmetic in Fourier space, we develop a general framework of moment transference principles, which convert Lebesgue data into data for some other measure.