Sets of bounded discrepancy for multi-dimensional irrational rotation

Hecke, Ostrowski and Kesten characterized the intervals on the circle
for which the ergodic sums of their indicator function, under an
irrational rotation, stay at a bounded distance from their integral
with respect to the Lebesgue measure on the circle.
In this talk I will discuss this phenomenon in multi-dimensional setting.
Based on joint work with Sigrid Grepstad.