Resource Bounded Randomness and Weakly Complete Problems Klaus Ambos­Spies, Sebastiaan A. Terwijn, Xizhong Zheng We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure. We concentrate on n^c­randomness (c >= 2) which corresponds to the polynomial time bounded (p­)measure of Lutz, and which is adequate for studying the internal and quantitative structure of E = DTIME(2^lin). However we will also comment on E_2 = DTIME(2^pol) and its corresponding (p_2­)measure. First we show that the class of n^c­random sets has p­measure 1. This provides a new, simplified approach to p­measure 1­results. Next we compare randomness with genericity, and we show that n^(c+1)­random sets are n^c­generic, whereas the converse fails. From the former we conclude that n^c­random sets are not p­btt­complete for E. Our technical main results describe the distribution of the n^c­random sets under p­m­reducibility. We show that every n^c­random set in E has n^k­random predecessors in E for any k >= 1, whereas the amount of randomness of the successors is bounded. We apply this result to answer a question raised by Lutz: We show that the class of weakly complete sets has measure 1 in E and that there are weakly complete problems which are not p­btt­complete for E.