Power monoids: A bridge between factorization theory and arithmetic combinatorics

Fan, Yushuang; Tringali, Salvatore

doi:10.1016/j.jalgebra.2018.07.010

Cited by 48 publications

(63 citation statements)

References 31 publications

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“…Alon [1] used probabilistic arguments combined with spectral techniques to improve earlier bounds by Green [8] on the maximal cardinality of subsets of a cyclic group of prime order that cannot be expressed as a sumset. Fan and Tringali [4] use tools from factorization theory to give (among other results) necessary and sufficient conditions for certain subsets of integers to be written as sumsets in more than one way. Selfridge and Straus [10] showed that the representation function r A (n) = |{(a, a ′ ) ∈ A × A : n = a + a ′ }| of a subset A in a field of characteristic zero determines the set.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Sidon set systems

2020

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A family A of k-subsets of {1, 2, . . . , N } is a Sidon system if the sumsets A+ B, A, B ∈ A are pairwise distinct. We show that the largest cardinality F k (N ) of a Sidon system of k-. More precise bounds on F k (N ) are obtained for k ≤ 3. We also obtain the threshold probability for a random system to be Sidon for k ≥ 2.This upper bound is tight for k = 2, that is, F 2 (N ) = 2N − 3. We believe that it is asymptotically sharp for any k ≥ 2, but we are only able to prove this for k = 2 and k = 3. For k ≥ 4, we prove that N k−1 is the right order of magnitude.Mathematics Subject Classification (2010): Primary 11B13; Secondary 05B10.

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Section: Introduction and Resultsmentioning

confidence: 99%

Sidon set systems

2020

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“…We end this subsection with a class of monoids stemming from additive combinatorics. [31,9,89]). For simplicity of presentation, we restrict ourselves to P fin (N 0 ) and to P fin,0 (N 0 ) consisting of all finite nonempty subsets of N 0 containing 0.…”

Section: 1mentioning

confidence: 99%

Factorization theory in commutative monoids

Geroldinger

Zhong

2020

Semigroup Forum

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This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

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“…We refer to [5, Remarks 2.17-2.20] for a critical comparison of these definitions with analogous ones from the literature on factorization theory: In particular, a weak transfer homomorphism in the sense of [1, Definition 2.1] is an essentially surjective equimorphism, by [5,Remark 2.19].…”

Section: A Focus On Systems Of Sets Of Lengthsmentioning

confidence: 99%

“…The subject developed out of algebraic number theory, and a turning point in its history has been the crucial observation, which can be traced back to the early work of F. Halter-Koch and A. Geroldinger in the area, that questions of non-unique factorization in integral domains are purely multiplicative in nature and, hence, can be conveniently rephrased in the language of monoids, with the latter providing "canonical models" of the phenomena under consideration that would not be available otherwise [10]. It is, however, only in recent years that fundamental aspects of factorization theory have been systematically extended to non-commutative or non-cancellative settings, see [2,9,5] and references therein. Notably, an impetus to these developments has come from a more profound comprehension of the interplay between factorization theory and arithmetic combinatorics, which is also the leitmotif of this paper.…”

Section: Introductionmentioning

confidence: 99%

Structural properties of subadditive families with applications to factorization theory

Tringali

2019

Isr. J. Math.

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Let H be a multiplicatively written monoid. Given k ∈ N + , we denote by U k the set of all ℓ ∈ N + such that a 1 · · · a k = b 1 · · · b ℓ for some atoms (or irreducible elements) a 1 , . . . , a k , b 1 , . . . , b ℓ ∈ H. The sets U k are one of the most fundamental invariants studied in the theory of non-unique factorization, and understanding their structure is a basic problem in the field: In particular, it is known that, in many cases of interest, these sets are almost arithmetic progressions with the same difference and bound for all large k, which is usually expressed by saying that H satisfies the Structure Theorem for Unions. The present paper improves the current state of the art on this problem.More precisely, we will show that, under mild assumptions on H, not only does the Structure Theorem for Unions hold, but there also exists µ ∈ N + such that, for every M ∈ N, the sequencesare µ-periodic from some point on. The result applies, for instance, to (the multiplicative monoid of) all commutative Krull domains (e.g., Dedekind domains) with finite class group; a variety of weakly Krull commutative domains (including all orders in number fields with finite elasticity); some maximal orders in central simple algebras over global fields; and all numerical monoids. Large parts of the proofs are worked out in a "purely additive model" (where no explicit reference to monoids or atoms is ever made), by inquiring into the properties of what we call a subadditive family, i.e., a collection L of subsets of N such that, for all L 1 , L 2 ∈ L , there is L ∈ L with L 1 + L 2 ⊆ L.

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Power monoids: A bridge between factorization theory and arithmetic combinatorics

Cited by 48 publications

References 31 publications

Sidon set systems

Sidon set systems

Factorization theory in commutative monoids

Structural properties of subadditive families with applications to factorization theory

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