Charlie Wilson

Some new results on hitting points infinitely often and limsup sets

We examine some fairly easily stated measure theory problems about covering a space with sets and seeing which points are hit infinitely often. That is if we fix the measures of the finite intersections, what can we say about the measure of the limsup. This will tie in to the Divergence Borel Cantelli Lemma and Erdos Chung inequality and show that there are limits to the use of such inequalities. We will go on to see that what we may feel is intuitively true and also what the current state of the art may indicate actually isn't- via some counterexamples that will be shown. The result, though a nice problem in isolation, does indeed have ramifications in many other areas of maths such as Probability theory, Diophantine approximation and Dynamical systems.