Exceptional directions for the Teichm\"{u}ller geodesic flow and Hausdorff dimension

We prove that for every flat surface \(\omega\), the Hausdorff dimension of the set of directions in which Teichm\"{u}ller geodesics starting from \(\omega\) exhibit a definite amount of deviation from the correct limit in Birkhoff's and Oseledets' Theorems is strictly less than \(1\). This theorem extends a result by Chaika and Eskin where they proved that such sets have measure \(0\). We also prove that the Hausdorff dimension of the directions in which Teichm\"{u}ller geodesics diverge on average in a stratum is bounded above by \(1/2\), strengthening a classical result due to Masur. Moreover, we show that the Hausdorff codimension of the set of non-weakly mixing IETs with permutation \((d, d-1, \dots, 1)\), where \(d\) is an odd number, is exactly \(1/2\) and strengthen a result by Avila and Leguil.