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account of lemma argument assertion follows bad terms base of G Bernoulli Shifts blocks called column of G Conditional entropy conditional expectation congruent copy construction contributes at least copy G copy of G defined definition of replicas disjoint Doctor of Philosophy double overlap e-levels entropy Ergodic Theory G columns G in G G v e gadgets G height of G hn-l integer interval invertible measure preserving K-automorphisms K-shifts law of large least eN bad lemma 9 lemma follows levels of G m-bad terms m+n m+n m+n)-block m+q+l measure of G measure preserving mapping multiple n-l)-blocks name of x Pairwise independence partition piece in G pth root replicas of G rigidity lemmas Section semi-bad terms sequence sigma-algebra SSI SSI Suppose tail field tier of G top of G total number union wlog zero-blocks