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A structure theorem for sets of small popular doubling

Volume 171 / 2015

Przemysław Mazur Acta Arithmetica 171 (2015), 221-239 MSC: Primary 11P70. DOI: 10.4064/aa171-3-2

Abstract

We prove that every set $A\subset\mathbb{Z}$ satisfying $\sum_{x}\min(1_A*1_A(x),t)\le (2+\delta)t|A|$ for $t$ and $\delta$ in suitable ranges 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 $A\subset\mathbb{N}$ satisfies $|\mathbb{N}\setminus(A+A)|\ge k$; specifically, we show that $\mathbb{P}(|\mathbb{N}\setminus(A+A)|\ge k)=\varTheta(2^{-k/2})$.

Authors

  • Przemysław MazurMathematical Institute
    Radcliffe Observatory Quarter
    Woodstock Road, Oxford OX2 6GG, United Kingdom
    e-mail

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