How well distributed are $\{i\rho\}_{i=1}^\infty$ mod 1?
Arrays of well distributed points are an important tool in Numerical Analysis. Irrational rotations play a central role in Ergodic Theory, Dynamical Systems, and Number Theory. Discrepancy (Pisot, Van Der Corput, 1930's) characterizes how evenly distributed a sequence of numbers in $[0,1)$ is. We study the discrepancy of $\{x_0+i\rho\}_{i=1}^n$.
The Birkhoff measure $\nu(\rho,n,z) dz$ associated to ${\rm frac}(x_0+i\rho)$ for $i=1$ to $n$ is the probability that $\sum_{i=1}^n[{{\rm frac}}(x_0+i\rho)-1/2]$ is in $[z,z+dz)$ if the distribution of $x_0$ is uniform on the circle.
New results: the graph of the Birkhoff measure $\nu(z)$ is a tile. If $n$ is a continued fraction denominator of $\rho$, then that graph is an isosceles trapezoid. The length of the support of $\nu$ equals the discrepancy (up to scaling).
We also give new and much more efficient proofs of two classical - but largely forgotten - results that allow one to compute the exact value of the discrepancy. We indicate some of these exact results.