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We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodic properties of translation flow than hyperbolic geometry.

Let $ R_α$ be an irrational rotation of the circle, and code the orbit of any point $ x$ by whether $ R_α^i(x) $ belongs to $ [0,α)$ or $ [α, 1)$ - this produces a Sturmian sequence. A point is undetermined at step $ j$ if its coding up to time $ j$ does not determine its coding at time $ j+1$. We prove a pair of results on the asymptotic frequency of a point being undetermined, for full measure sets of $ α$ and $ x$.

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