Iterated sumsets and setpartitions

Grynkiewicz, David J.

doi:10.1007/s11139-019-00155-y

Cited by 1 publication

(3 citation statements)

References 40 publications

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“…In Section 3, we will also derive some additional strengthenings of Theorem 1.1 in the case n ≥ 1 2 |S ′ | is very large. In particular, we will achieve the same strengthened conclusions recently guaranteed in [38] under a different n is large assumption (Theorem 3.2). This, in turn, will allow us to derive additional information for S, in particular, when |S| = 2n with |Σ n (S)| ≤ n+1 and h(S) ≤ n (Theorem 3.3).…”

Section: Note |φ H (S ′ )| =supporting

confidence: 78%

“…We now derive our strengthening of Theorem 1.1 for large n, mirroring the main result from [38] (which obtained the same conclusion assuming n is large with respect to the exponent). Theorem 3.2.…”

Section: Partitioning Results For Large Nsupporting

confidence: 54%

“…Kneser's Theorem is the fundamental lower bound for sumsets in an abelian group. Combining it with Theorem A (applied modulo H) yields the analogous result for sequence subsums [38].…”

Section: Introductionmentioning

confidence: 61%

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Representing Sequence Subsums as Sumsets of Near Equal Sized Sets

Grynkiewicz¹

2019

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For a sequence S of terms from an abelian group G of length |S|, let Σn(S) denote the set of all elements that can be represented as the sum of terms in some n-term subsequence of S. When the subsum set is very small, |Σn(S)| ≤ |S| − n + 1, it is known that the terms of S can be partitioned into n nonempty sets A1, . . . , An ⊆ G such that Σn(S) = A1 + . . . + An. Moreover, if the upper bound is strict, then |Ai \ Z| ≤ 1 for all i, where Z = n i=1 (Ai + H) and H = {g ∈ G : g + Σn(S) = Σn(S)} is the stabilizer of Σn(S). This allows structural results for sumsets to be used to study the subsum set Σn(S) and is one of the two main ways to derive the natural subsum analog of Kneser's Theorem for sumsets. In this paper, we show that such a partitioning can be achieved with sets Ai of as near equal a size as possible, so ⌊ |S| n ⌋ ≤ |Ai| ≤ ⌈ |S| n ⌉ for all i, apart from one highly structured counterexample when |Σn(S)| = |S| − n + 1 with n = 2. The added information of knowing the sets Ai are of near equal size can be of use when applying the aforementioned partitioning result, or when applying sumset results to study Σn(S) (e.g., [20]). We also give an extension increasing the flexibility of the aforementioned partitioning result and prove some stronger results when n ≥ 1 2 |S| is very large.

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Section: Note |φ H (S ′ )| =supporting

confidence: 78%

Section: Partitioning Results For Large Nsupporting

confidence: 54%