Counting odd cycles in locally dense graphs

We prove that for any given @e>0 and d@?[0,1], every sufficiently large (@e,d)-dense graph G contains for each odd integer r at least (d^r-@e)|V(G)|^r cycles of length r. Here, being (@e,d)-dense means that every set X containing at least @e|V(G)| vertices spans at least 12@?d|X|^2 edges, and what we really count is the number of homomorphisms from an r-cycle into G. The result addresses a question of Y. Kohayakawa, B. Nagle, V. Rodl, and M. Schacht.