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Random Cayley graphs and random sumsets
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abstract
We prove that any finite abelian group $G$ contains a collection of not too many subsets with a special structure, so that for every subset $A$ of $G$ with a small doubling, there is a member $F$ of the collection that is fully contained in the sumset $A+A$ and is not much smaller than it. Using this result we obtain improved bounds for the problem of estimating the typical independence number of sparse random Cayley or Cayley-sum graphs, and for the problem of estimating the smallest size of a subset of $G$ which is not a sumset. We also obtain tight bounds for the typical maximum length of an arithmetic progression in the sumset of a sparse random subset of $G$.
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Dense sets without large sumsets
A δ-dense random subset of [n] almost surely contains no sumset A+B unless one side has size below about 3 log n/log(1/δ), matching the known lower bound up to factor 3.
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