Anticoncentration versus the number of subset sums

01/19/2021

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by   Vishesh Jain, et al.

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Let w⃗ = (w_1,…, w_n) ∈ℝ^n. We show that for any n^-2≤ϵ≤ 1, if #{ξ⃗∈{0,1}^n: ⟨ξ⃗, w⃗⟩ = r}≥ 2^-ϵ n· 2^n for some r ∈ℝ, then #{⟨ξ⃗, w⃗⟩ : ξ⃗∈{0,1}^n}≤ 2^O(√(ϵ)n). This exponentially improves a recent result of Nederlof, Pawlewicz, Swennenhuis, and Węgrzycki and leads to a similar improvement in the parameterized (by the number of bins) runtime of bin packing.