Week 5
Ex 1: Show that if vol(C)>k2^d, then it contains at least 2k gridpoints.
Ex 2: For any 0<t1, t2<1 and N natural, exists m1, m2, n<=N natural: |ti-mi/n|<1/nsqrt(N).
Week 6
Ex1: Given n lines in the plane, show that you can properly color the cells formed by these lines black and white. (In a proper coloring adjacent cells don't have the same color. Cells that share an edge are adjacent).
Ex2: Prove the following 'circular' version of cutting lemma: Given n circles, C, in the plane, let S be a random subset (selection with replacement) of s=Crlogn circles. Draw a vertical line upwards and downwards from each vertex (point of intersection of circles). Show that there are O(s^2) regions/circular trapezoids. Show that if C is a sufficiently large constant, then with a positive probability S intersects all possible trapezoids with more than n/r circles in their interior.