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

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