Resumen
* HOW WELL DISTRIBUTED ARE {iρ}i=1∞ MOD 1 ? Arrays of well distributed points are an important tool in Num. Anal. Irrational rotations play a central role in Erg Th, Dyn Sys, and Num Th.
* Discrepancy (Pisot, Van Der Corput, 1930’s) characterizes how evenly distributed a sequence of numbers in [0,1) is. We study the discrepancy of {x0+iρ}i=1∞.
* The Birkhoff measure ν(ρ,n,z)dz associated to frac(x0+iρ) for i=1 to n is the probability that Σi=1n[frac(x0+iρ)−1/2] is in z+dz if the distribution of x0 is uniform on the circle.
* New results: the graph of the Birkhoff measure ν(z) is a tile. If n is a continued fraction denominator of ρ, then that graph is an isosceles trapezoid. The length of the support of ν(z) 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.
Ponente
Dr. J. J. P. Veerman
Math/Stat, Portland State University