A structure theorem for sets of small popular doubling

Przemysław Mazur

Acta Arithmetica, Tome 168 (2015), p. 221-239 / Harvested from The Polish Digital Mathematics Library

Résumé

We prove that every set A ⊂ ℤ satisfying $\sum_{x} m i n (1_{A} * 1_{A} (x), t) \leq (2 + δ) t | A |$ for t and δ 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 ⊂ ℕ satisfies |ℕ∖(A+A)| ≥ k; specifically, we show that $ℙ (| ℕ ∖ (A + A) | \geq k) = Θ (2^{- k / 2})$ .

Publié le : 2015-01-01

Zbl 06498808

EUDML-ID : urn:eudml:doc:279419

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     title = {A structure theorem for sets of small popular doubling},
     journal = {Acta Arithmetica},
     volume = {168},
     year = {2015},
     pages = {221-239},
     zbl = {06498808},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa171-3-2}
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Przemysław Mazur. A structure theorem for sets of small popular doubling. Acta Arithmetica, Tome 168 (2015) pp. 221-239. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa171-3-2/