On the sharpness of the R\"ussmann estimates

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.