Maximum subsets of (0,1] with no solutions to x+y = kz

Chung, Fan; Goldwasser, John L.

doi:10.37236/1225

Cited by 8 publications

(17 citation statements)

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“…Chung and Goldwasser also initiated the study of k-sum-free sets in the continuous setting. In particular, in [8] 2 The term k-sum-free set is used for instance in [1]. These sets should not be confused with sets free of solutions to the equation a 1 + · · · + a k = b, which have also been called k-sum-free sets (see [19]).…”

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

confidence: 99%

“…The first application concerns the problem of determining the supremum of measures of Borel sets A ⊂ T such that the cartesian power A 3 contains no triple (x, y, z) solving the equation x + y = kz, where k ≥ 3 is a fixed integer. This is an analogue in T of a problem which goes back to Erdős (see [7]) and which has been treated in several works, first in the integer setting (see in particular [1,7]) and then also in the continuous setting of an interval in R [8,21,23]. The above-mentioned supremum is seen to be at most 1/3 by a simple application of Raikov's inequality from [24] (see also [20,Theorem 1] Acknowledgements.…”

Section: Introductionmentioning

confidence: 99%

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

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On sets with small sumset in the circle

Candela

Roton

2018

The Quarterly Journal of Mathematics

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“…These authors additionally proved an inverse theorem giving the structure of a k-sum-free sets of this size : such sets have to be close from the set composed of the three above-mentioned intervals. In fact, Chung and Golwasser managed to predict the maximal size of a k-sum-free set of integers less than n by studying the continuous analog of the problem in [3]; in other words by introducing the study of k-sum-free subsets of reals number selected from [0, 1]. Indeed, a k-sum-free subset of [0, 1] leads, after a suitable dilation, to a k-sum-free set of integers (but it is important to notice, this set will be mandatorily -in the typology mentioned above -of a combinatorial nature).…”

Section: Introductionmentioning

confidence: 99%

“…We thus arrive to the question of determining the maximal Lebesgue measure -denoted thereafter µ -of a subset of [0, 1] having no solution to the equation x + y = kz. The case k = 1 is easy and, as mentioned above, the cases k ≥ 4 were solved in [3]. However, the case k = 3 was left open and remained the only one for which the optimal asymptotic density was unknown.…”

Section: Introductionmentioning

confidence: 99%

“…These seven sets are 3-sum-free. The quite precise following conjecture was then formulated in [3] (and proposed again later in [7] after several computeraided checks): Recently, Matolcsi and Ruzsa [7] made the first breakthrough towards the first part of this conjecture by showing the following theorem. This result, the first one to prove a strictly less than 0.5 upper bound for 3-sum-free subsets of [0, 1], is very noticeable because it shows in particular that in the case of 3-sumfree sets, contrary to what happens in the other cases, the maximal size of such a subset in [0, 1] is not the analog of that of a k-sum-free subset of integers (let us recall that such a set has a density 1/2).…”

Section: Introductionmentioning

confidence: 99%

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Maximal sets with no solution to x+y=3z

Plagne

Roton

2015

Combinatorica

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In this paper, we are interested in a generalization of the notion of sum-free sets. We address a conjecture first made in the 90s by Chung and Goldwasser. Recently, after some computer checks, this conjecture was formulated again by Matolcsi and Ruzsa, who made a first significant step towards it. Here, we prove the full conjecture by giving an optimal upper bound for the Lebesgue measure of a 3-sum-free subset A of [0, 1], that is, a set containing no solution to the equation x + y = 3z where x, y and z are restricted to belong to A. We then address the inverse problem and characterize precisely, among all sets with that property, those attaining the maximal possible measure.

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