# On the sharpness of the R\"ussmann estimates

## TL;DR

This paper compares classical Rüssmann estimates for solving linear difference equations in KAM theory with computer-assisted estimates, analyzing their accuracy and potential for improvement through experiments.

## Contribution

It introduces a systematic comparison between traditional and computer-assisted estimates, highlighting sources of overestimation and suggesting directions for refinement.

## Key findings

- Computer-assisted estimates are closer to actual solutions.
- Classical Rüssmann estimates tend to overestimate the solution norm.
- Experiments identify key sources of overestimation in traditional bounds.

## Abstract

Estimating the norm of the solution of the linear difference equation $u(\theta)-u(\theta+\omega)=v(\theta)$ plays a fundamental role in KAM theory. Optimal (in certain sense) estimates for the solution of this equation were provided by R\"ussmann in the mid 70's. The aim of this paper is to compare the sharpness of these classic estimates with more specific estimates obtained with the help of the computer. We perform several experiments to quantify the improvement obtained when using computer assisted estimates. By comparing these estimates with the actual norm of the solution, we can analyze the different sources of overestimation, thus encouraging future improvements.