Approximating Sumset Size
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Given a subset A of the n-dimensional Boolean hypercube 𝔽_2^n, the sumset A+A is the set {a+a': a, a' ∈ A} where addition is in 𝔽_2^n. Sumsets play an important role in additive combinatorics, where they feature in many central results of the field. The main result of this paper is a sublinear-time algorithm for the problem of sumset size estimation. In more detail, our algorithm is given oracle access to (the indicator function of) an arbitrary A ⊆𝔽_2^n and an accuracy parameter ϵ > 0, and with high probability it outputs a value 0 ≤ v ≤ 1 that is ±ϵ-close to Vol(A' + A') for some perturbation A' ⊆ A of A satisfying Vol(A ∖ A') ≤ϵ. It is easy to see that without the relaxation of dealing with A' rather than A, any algorithm for estimating Vol(A+A) to any nontrivial accuracy must make 2^Ω(n) queries. In contrast, we give an algorithm whose query complexity depends only on ϵ and is completely independent of the ambient dimension n.
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