Algebraic Proof for the Geometric Structure of Sumsets

Lee, Jae Woo

doi:10.1515/integ.2011.034

Cited by 6 publications

(6 citation statements)

References 9 publications

(7 reference statements)

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“…Iterated Minkowski sums of a finite positively generating set of Z d are "roughly" equal to the integer points in the dilated convex hull. A proof of this fact appears in [Lee11]. We provide a slightly different short proof based on the well known Shapley-Folkman Lemma (see for instance [Cas75]).…”

Section: The Geometry Of Horoballs In Z Dmentioning

confidence: 98%

Iterated Minkowski sums, horoballs and north-south dynamics

Epperlein¹,

Meyerovitch²

2020

Preprint

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Given a finite generating set A for a group Γ, we study the map W → W A as a topological dynamical system -a continuous self-map of the compact metrizable space of subsets of Γ. If the set A generates Γ as a semigroup and contains the identity, there are precisely two fixed points, one of which is attracting. This supports the initial impression that the dynamics of this map is rather trivial. Indeed, at least when Γ = Z d and A ⊆ Z d a finite positively generating set containing the natural invertible extension of the map W → W + A is always topologically conjugate to the unique "northsouth" dynamics on the Cantor set. In contrast to this, we show that various natural "geometric" properties of the finitely generated group (Γ, A) can be recovered from the dynamics of this map, in particular, the growth type and amenability of Γ. When Γ = Z d , we show that the volume of the convex hull of the generating set A is also an invariant of topological conjugacy. Our study introduces, utilizes and develops a certain convexity structure on subsets of the group Γ, related to a new concept which we call the sheltered hull of a set. We also relate this study to the structure of horoballs in finitely generated groups, focusing on the abelian case.

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Section: The Geometry Of Horoballs In Z Dmentioning

confidence: 98%

Iterated Minkowski sums, horoballs and north-south dynamics

Epperlein¹,

Meyerovitch²

2020

Preprint

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“…, w n+1 ], we get the claim. The last equality follows from the fact that in [23] and [24] it is established that if Z(A − A) has maximal rank, then the leading coefficient of the polynomial p…”

Section: Sumsets and Monomial Projections Of Veronese Varietiesmentioning

confidence: 99%

“…Notwithstanding, Khovanskii's result sheds no light on the polynomial p A (t), excepting the leading coefficient, nor the phase transition of ϕ A (t). In view of these considerations, many contributions have been made to these topics as one can see, for instance, in [4,8,9,13,14,15,24,27,28,29].…”

Section: Introductionmentioning

confidence: 99%

Sumsets and Veronese varieties

Colarte-Gómez¹,

Elias²,

Miró‐Roig³

2022

Preprint

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In this paper, to any subset A ⊂ Z n we explicitly associate a unique monomial projection Y n,d A of a Veronese variety, whose Hilbert function coincides with the cardinality of the t-fold sumsets tA. This link allows us to tackle the classical problem of determining the polynomial pA ∈ Q[t] such that |tA| = pA(t) for all t ≥ t0 and the minimum integer n0(A) ≤ t0 for which this condition is satisfied, i.e. the so-called phase transition of |tA|. We use the Castelnuovo-Mumford regularity and the geometry of Y n,d A to describe the polynomial pA(t) and to derive new bounds for n0(A) under some technical assumptions on the convex hull of A; and vice versa we apply the theory of sumsets to obtain geometric information of the varieties Y n,d A .

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“…what "sufficiently large" means). There have been other proofs of Khovanskii's theorem since, including a geometric proof (which also patches an error in Khovanskii's original paper) by Lee [8] and a purely combinatorial proof by Nathanson and Ruzsa [11,12], but to our knowledge no effective version of Khovanskii's theorem is known for subsets of Z d for any d > 1. In this paper we give a different approach to Khovanskii's theorem that yields more information than previous approaches about the structure of the polynomial and where the phase transition occurs.…”

Section: Introductionmentioning

confidence: 99%

Khovanskii's theorem and effective results on sumset structure

Curran¹,

Goldmakher²

2020

Preprint

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A remarkable theorem due to Khovanskii asserts that for any finite subset A of an abelian group, the cardinality of the h-fold sumset hA grows like a polynomial for all sufficiently large h. Currently, neither the polynomial nor what sufficiently large means are understood. In this paper we obtain an effective version of Khovanskii's theorem for any A ⊂ Z d whose convex hull is a simplex; previously, such results were only available for d = 1. Our approach gives information about not just the cardinality of hA, but also its structure, and we prove two effective theorems describing hA as a set: one answering a recent question posed by Granville and Shakan, the other a Brion-type formula that provides a compact description of hA for all large h. As a further illustration of our approach, we derive a completely explicit formula for |hA| whenever A ⊂ Z d consists of d+2 points.

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Algebraic Proof for the Geometric Structure of Sumsets

Cited by 6 publications

References 9 publications

Iterated Minkowski sums, horoballs and north-south dynamics

Iterated Minkowski sums, horoballs and north-south dynamics

Sumsets and Veronese varieties

Khovanskii's theorem and effective results on sumset structure

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