The Structure of Maximum Subsets of $\{1,\ldots,n\}$ with No Solutions to $a+b = kc$

Baltz, Andreas; Hegarty, Peter; Knape, Jonas; Larsson, Urban; Schoen, Tomasz

doi:10.37236/1916

Cited by 14 publications

(41 citation statements)

References 4 publications

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“…This set has asymptotic density 1 m+2 , and is still the greatest known example for m ≤ 7. However, for larger values of m, a construction of Tomasz Schoen (personal communication), presented in this paper in Lemma 3.1 with his kind permission, yields an improved lower bound of the form d m ≥ 1 8 − o m→∞ (1). We summarize these results as follows.…”

Section: Introductionmentioning

confidence: 65%

“…For a positive integer m, a subset A of an abelian group is said to be m-sum-free if there is no triple (x, y, z) ∈ A 3 satisfying x + y = mz. These sets have been studied in numerous works in arithmetic combinatorics, including various types of abelian group settings [1,6,7,8,13] (see also [4,Section 3] for an overview of this topic). In Z/pZ, a central goal concerning these sets is to estimate the quantity…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On sets with small sumset and m-sum-free sets in Z/pZ

Candela¹,

González-Sánchez²,

Grynkiewicz³

2019

Preprint

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This paper makes progress toward the 3k − 4 conjecture in groups Z/pZ for p prime. The conjecture states that if A is a nonempty subset of Z/pZ satisfying 2A = Z/pZ and |2A| = 2|A| + r ≤ min{3|A| − 4, p − r − 3}, then A is covered by an arithmetic progression of size at most |A| + r + 1. A theorem of Serra and Zémor proves the conjecture provided that r ≤ 0.0001|A|, without any additional constraint on |A|. Subject to the mild constraint |2A| ≤ 3p/4 (which is optimal in a sense explained in the paper), our first main result improves the bound on r, allowing r ≤ 0.1368|A|. We also prove a variant that further improves this bound on r provided A is sufficiently dense. We then apply this variant to give a new upper bound for the maximal density of m-sum-free sets in Z/pZ, i.e., sets A having no solution (x, y, z) ∈ A 3 to the equation x + y = mz, where m ≥ 3 is a fixed integer. The previous best upper bound for this maximal density was 1/3.0001 (using the Serra-Zémor Theorem). We improve this to 1/3.1955. We also present a construction of Schoen that yields a lower bound for this maximal density of the form 1/8 − om(1)p→∞.

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Section: Introductionmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

On sets with small sumset and m-sum-free sets in Z/pZ

Candela¹,

González-Sánchez²,

Grynkiewicz³

2019

Preprint

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“…They additionally conjectured that this was the actual exact asymptotic maximal value. This conjecture was finally settled by Baltz, Hegarty, Knape, Larsson and Schoen in [1]. 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.…”

Section: Introductionmentioning

confidence: 94%

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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“…Note that the proposition implies in particular that λ 0 = ρ for any equation in two variables. For three variables things get more interesting and a number of papers have been entirely devoted to this situation, see [1] [2]…”

Section: Setmentioning

confidence: 99%

“…[4] [6] [7] plus the multitude of papers on sum-free sets, of which the most directly relevant is probably [3]. The combined results of [1], [2] and [3] give, in principle, a complete classification of the extremal L-avoiding subsets of [1, n], for every n > 0, and L : x + y = cz for any c = 2. Of particular interest for us are the results of [1].…”

Section: Setmentioning

confidence: 99%