Large sets with small doubling modulo p are well covered by an arithmetic progression

Serra, Oriol; Zémor, Gilles

doi:10.5802/aif.2482

Cited by 18 publications

(44 citation statements)

References 16 publications

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“…The greatest value of ε currently available here is the one provided by Serra and Zémor in [30], namely ε = 10 −4 . This gives us d k (T) ≤ 1 3+10 −4 for all k ≥ 3.…”

Section: Application To K-sum-free Sets In Tmentioning

confidence: 99%

On sets with small sumset in the circle

Candela

Roton

2018

The Quarterly Journal of Mathematics

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We prove results on the structure of a subset of the circle group having positive inner Haar measure and doubling constant close to the minimum. These results go toward a continuous analogue in the circle of Freiman's 3k − 4 theorem from the integer setting. An analogue of this theorem in Z p has been pursued extensively, and we use some recent results in this direction. For instance, obtaining a continuous analogue of a result of Serra and Zémor, we prove that if a subset A of the circle is not too large and has doubling constant at most 2 + ε with ε < 10 −4 , then for some integer n > 0 the dilate n · A is included in an interval in which it has density at least 1/(1 + ε). Our arguments yield other variants of this result as well, notably a version for two sets which makes progress toward a conjecture of Bilu. We include two applications of these results.The first is a new upper bound on the size of k-sum-free sets in the circle and in Z p . The second gives structural information on subsets of R of doubling constant at most 3 + ε.PABLO CANDELA AND ANNE DE ROTON homomorphism x → n x (for A ⊂ Z p and n ∈ Z p we also use n · A to denote the image of A under x → n x). The conclusion of Conjecture 1.1 can be rephrased as follows: there exists n ∈ Z p \ {0} and an interval I ⊂ Z p such that n · A ⊂ I and |I| ≤ |A| + r.Freiman's 3k − 4 theorem has an extension applicable to two possibly different sets A, B [18,31]. A Z p -analogue of this extension has also been proposed, namely the so-called r-critical pair conjecture. A version of this conjecture appeared 1 in [17] and was proved for small sets in [3,15]. We recall the following more recent version [16, Conjecture 19.2].Conjecture 1.2. Let p be a prime, let r be a non-negative integer, and let A, B be nonempty subsets of Z p with |A| ≥ |B| and satisfying |A + B| = |A| + |B| + r − 1 ≤ 1 2 (p + |A| + |B|) − 2, and r ≤ |B| − 3.(1)

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“…The greatest value of ε currently available here is the one provided by Serra and Zémor in [30], namely ε = 10 −4 . This gives us d k (T) ≤ 1 3+10 −4 for all k ≥ 3.…”

Section: Application To K-sum-free Sets In Tmentioning

confidence: 99%

On sets with small sumset in the circle

Candela

Roton

2018

The Quarterly Journal of Mathematics

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“…Previous inverse theorems for sumsets focused on making the value of ǫ ′ as large as possible, even at the expense of requiring |A| to be small. Serra and Zémor [36], on the other hand, proved the above result allowing |A| to be as large as possible, at the expense of requiring ǫ ′ to be small. |A…”

Section: The Isoperimetric Methodsmentioning

confidence: 83%

“…The following is the main result from the isoperimetric method that we will use, and it was proven by Serra and Zémor [36,Theorem 3]. Previous inverse theorems for sumsets focused on making the value of ǫ ′ as large as possible, even at the expense of requiring |A| to be small.…”

Section: The Isoperimetric Methodsmentioning

confidence: 94%

“…To prove our main result (Theorem 1.6), we will combine ideas from the rectification method with a strong new result due to Serra and Zémor [36] (see Subsection 4.2 for a discussion of the Serra-Zémor result). The first step in our proof, which we will carry out in the next section, is to reduce the study of restricted sumsets to non-restricted sumsets.…”

Section: A Combinatorial Approachmentioning

confidence: 99%

“…[36] There exist positive numbers p 0 and ǫ ′ such that for all primes p > p 0 , any subset A of Z/pZ such that(i) |A + A| < (2 + ǫ ′ ) |A| and (ii) m = |A + A| − 2 |A| satisfies m min{|A| − 4, p − |A + A| − 3}is contained in an arithmetic progression of at most |A| + m + 1 terms. Furthermore, one can take ǫ ′ = 10 −4 and p 0 = 2 94 .…”

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confidence: 99%

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The Inverse Erdős-Heilbronn Problem

Wood

2009

Electron. J. Combin.

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The famous Erdős-Heilbronn conjecture (first proved by Dias da Silva and Hamidoune in 1994) asserts that if $A$ is a subset of ${\Bbb Z}/p{\Bbb Z}$, the cyclic group of the integers modulo a prime $p$, then $|A\widehat{+}A| \ge \min\{2|A| -3,p\}. $ The bound is sharp, as is shown by choosing $A$ to be an arithmetic progression. A natural inverse result was proven by Karolyi in 2005: if $A\subset {\Bbb Z}/p{\Bbb Z}$ contains at least 5 elements and $|A\widehat{+}A| \le 2|A| -3 < p$, then $A$ must be an arithmetic progression. We consider a large prime $p$ and investigate the following more general question: what is the structure of sets $A \subset {\Bbb Z}/p{\Bbb Z}$ such that $|A\widehat{+}A| \le (2+\epsilon)|A|$? Our main result is an asymptotically complete answer to this question: there exists a function $\delta(p) = o(1)$ such that if $200 < |A| \le (1-\epsilon')p/2$ and if $|A\widehat{+}A| \le (2+\epsilon)|A|$, where $\epsilon' -\epsilon \ge \delta > 0$, then $A$ is contained in an arithmetic progression of length $|A\widehat{+}A| -|A| + 3$. With the extra assumption that $|A| \le ({1\over 2}- {1\over\log^c p} ) p$, our main result has Dias da Silva and Hamidoune's theorem and Karolyi's theorem as corollaries, and thus, our main result provides purely combinatorial proofs for the Erdős-Heilbronn conjecture and an inverse Erdős-Heilbronn theorem.

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Compression, Complements and the 3k−4 Theorem

Grynkiewicz

2013

Structural Additive Theory

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The 3k − 4 Theorem for Z asserts that, if A, B ⊆ Z are finite, nonempty subsets with |A| ≥ |B| and |A + B| = |A| + |B| + r < |A| + 2|B| − 3, then there exist arithmetic progressions PA and PB of common difference such that X ⊆ PX with |PX | ≤ |X| + r + 1 for all X ∈ {A, B}. There is much partial progress extending this result to Z/pZ with p ≥ 2 prime. Here, among other results, we begin by showing that, if A, B ⊆ G = Z/pZ are nonempty subsets with |A| ≥ |B|, A + B = G, |A + B| = |A| + |B| + r ≤ |A| + 1.0527|B| − 3, and |A + B| ≤ |A| + |B| − 9(r + 3), then there exist arithmetic progressions PA, PB and PC of common difference such that X ⊆ PX with |PX | ≤ |X| + r + 1 for all X ∈ {A, B, C}, where C = − G \ (A + B). This gives a rare high density version of the 3k − 4 Theorem for general sumsets A + B and is the first instance with tangible (rather than effectively existential) values for the constants for general sumsets A + B without also imposing additional constraints on the relative sizes of |A| and |B|. The ideal conjectured density restriction under which a version of the 3k − 4 Theorem modulo p is expected is |A + B| ≤ p − (r + 3). In part by utilizing the above result as well as several other recent advances, we are able to extend methods of Serra and Zémor to give a version valid under this ideal density constraint. We show that, if A, B ⊆ G = Z/pZ are nonempty subsets with |A| ≥ |B|, A + B = G, |A + B| = |A| + |B| + r ≤ |A| + 1.01|B| − 3, and |A + B| ≤ |A| + |B| − (r + 3), then there exist arithmetic progressions PA, PB and PC of common difference such that X ⊆ PX with |PX | ≤ |X| + r + 1 for all X ∈ {A, B, C}, where C = − G \ (A + B). This notably improves upon the original result of Serra and Zémor [31], who treated the case A + A, required p be sufficiently large, and needed the much more restrictive small doubling hypothesis |A + A| ≤ |A| + 1.0001|A|.

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Large sets with small doubling modulo p are well covered by an arithmetic progression

Cited by 18 publications

References 16 publications

On sets with small sumset in the circle

On sets with small sumset in the circle

The Inverse Erdős-Heilbronn Problem

Compression, Complements and the 3k−4 Theorem

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