Balanced supersaturation for some degenerate hypergraphs

A classical theorem of Simonovits from the 1980s asserts that every graph (Formula presented.) satisfying (Formula presented.) must contain (Formula presented.) copies of (Formula presented.). Recently, Morris and Saxton established a balanced version of Simonovits' theorem, showing that such (Formula presented.) has (Formula presented.) copies of (Formula presented.), which are “uniformly distributed” over the edges of (Formula presented.). Moreover, they used this result to obtain a sharp bound on the number of (Formula presented.) -free graphs via the method of hypergraph containers. In this article, we generalise Morris–Saxton's results for even cycles to (Formula presented.) -graphs. We also prove analogous results for complete (Formula presented.) -partite (Formula presented.) -graphs.