A study in quantitative equidistribution on the unit square

The distributional properties of the translation flow on the unit square have been considered in different fields of mathematics, including algebraic geometry and discrepancy theory. One method to quantify equidistribution is to compare the error between the actual time the translation flow spent in specific sets $E \subset [0,1]^2$ to the expected time. In this article, we prove that when $E$ is in the algebra generated by convex sets the error is of order at most $\log(T)^{1+\varepsilon}$ for all but countably many directions. Whenever the direction is badly approximable the bound can be sharpened to $\log(T)^{1/2+\varepsilon}$. The error estimates we produce are smaller than for general measurable sets as proved by Beck, while our class of examples is larger than in the work of Grepstad-Larcher who obtained the bounded remainder property for their sets. Our proof relies on the duality between local convexity of the boundary and regularity of sections of the flow.