Research paperOn the sharpness of the Rüssmann estimates

- We quantify the improvement of classic Rüssmann estimates using computer assisted estimates.
- We analyze the different sources of overestimation of the actual norm of the solution.
- We consider suitably chosen test functions to capture the behavior of general functions.
- We consider the dependence on the different parameters involved, in particular the frequency.

Estimating the norm of the solution of the linear difference equation u(θ)−u(θ+ω)=v(θ) plays a fundamental role in KAM theory. Optimal (in certain sense) estimates for the solution of this equation were provided by Rüssmann in the mid 70's. The aim of this paper is to compare the sharpness of these classical 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.