Abstract: Suppose \alpha is an irrational number and let 
        D_N(\alpha)=#{1\leq n\leq N: n\alpha\mod 1 is in [0,1/2]}-N/2 .
This is a sequence of half-integers. I will characterize the \alpha of bounded type for which D_N(\alpha) is "equidistributed," in the sense that for any half integers x,y
        #{1            as M tends to infinity.
Corollary: For many \alpha (e.g. the square root of two), D_N(\alpha) is **not** equidistributed.
This is a new form of  ``irregularity in uniform distribution."
(Joint work with Dima Dolgopyat)