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Abstract A finite groupGis calledC-quasirandom (by Gowers) if all non-trivial irreducible complex representations ofGhave dimension at leastC. For any unit$$\ell ^{2}$$ function on a finite group we associate thequantum probability measureon the group given by the absolute value squared of the function. We show that if a group is highly quasirandom, in the above sense, then any Cayley graph of this group has an orthonormal eigenbasis of the adjacency operator such that the quantum probability measures of the eigenfunctions put close to the correct proportion of their mass on suitably selected subsets of the group that are not too small.more » « less
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Abstract We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma , states that for each $$\epsilon>0$$ there exists M such that every triangle-free graph G has an $$\epsilon$$ -approximate homomorphism to a triangle-free graph F on at most M vertices (here an $$\epsilon$$ -approximate homomorphism is a map $$V(G) \to V(F)$$ where all but at most $$\epsilon \left\lvert{V(G)}\right\rvert^2$$ edges of G are mapped to edges of F ). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in $$\epsilon^{-1}$$ . We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal.more » « less
