A subset S of the Boolean hypercube F₂ⁿ is a sumset if S = A+A = + b \ | \ a, b∈ A\ for some A ⊆ F₂ⁿ. We prove that the number of sumsets in F₂ⁿ is asymptotically (2ⁿ-1)2^2ⁿ⁻¹. Furthermore, we show that the family of sumsets in F₂ⁿ is almost identical to the family of all subsets of F₂ⁿ that contain a complete linear subspace of co-dimension $1$.
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Alon et al. (2024) studied this question.
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