A Constructive Proof of Ky Fan\u27s Generalization of Tucker\u27s Lemma

We present a proof of Ky Fan\u27s combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fan\u27s lemma to allow for triangulations of Sn that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tucker\u27s lemma that holds for a more general class of triangulations than the usual version

International Social Security Review, Vol. 60, 2007

Given n vectors {i} ∈ [0, 1)d, consider a random walk on the d-dimensional torus d = ℝd/ℤd generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*k) ≥ C1k−n/2, where C1 = C(n, d) is a constant. If the vectors are badly approximated by rationals (in a sense we will define), then D(Q*k) ≤ C2k−n/2d for C2 = C(n, d, j) a constant