A structure theorem for sets of small popular doubling

2015/06/01 by Przemysław Mazur, Mazur, Przemysław
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1506.00445

arxiv created 2015/06/01 · arxiv updated 2015/06/02

Abstract

In this paper we prove that every set A⊂ℤ satisfying the inequality ∑_xmin(1_A*1_A(x),t)≤(2+δ)t|A| for t and δ in suitable ranges, then A must be very close to an arithmetic progression. We use this result to improve the estimates of Green and Morris for the probability that a random subset A⊂ℕ satisfies |ℕ∖(A+A)|≥ k; specifically we show that ℙ(|ℕ∖(A+A)|≥ k)=Θ(2^-k/2).

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