On a partition analog of the Cauchy-Davenport theorem

Abstract

Let G be a finite abelian group, and let n be a positive integer. From the Cauchy-Davenport Theorem it follows that if G is a cyclic group of prime order, then any collection of n subsets A1, A2, . . . , An of G satisfiesM. Kneser generalized the Cauchy-Davenport Theorem for any abelian group. In this paper, we prove a sequence-partition analog of the Cauchy-Davenport Theorem along the lines of Kneser's Theorem. A particular case of our theorem was proved by J. E. Olson in the context of the Erdős-Ginzburg-Ziv Theorem.

A Weighted Erdős-Ginzburg-Ziv Theorem

“…Finally, we remark that the proof when w i = 1 for all i of the following corollary to Theorem 2.2 likewise goes through in the more general weighted case (Theorem 2.7 in [21] and Corollary 1 in [23]). Theorem 2.3.…”

“…, it follows that the proof given in [23] goes through in the more general weighted case with no further modifications other than to insert the weights w i at appropriate points in the proof. Theorem 2.2.…”

“…Hence, since H ka ≤ H a , it follows that |φ a (w j A j )| < |w j A j | for some index j with 2 ≤ j ≤ l − 1. Thus, since by re-indexing we may assume j = l − 1, it follows that (3) holds with j = l − 1, which in view of (4) contradicts the minimality of l. Theorem 1.1 has a much more potent generalization, provided every w i is relatively prime to k. The following theorem, which implies Theorem 1.1 when every w i is relatively prime to k, was proved in the case w i = 1 for all i in [23]. However, noting for w i relatively prime to k that w i b ∈ w i A i if and only if b ∈ A i , and allowing for re-indexing of the weights…”

“…We can now begin the proof of Theorem 1.1, which follows ideas from the proof of a recent composite analog of the Cauchy-Davenport Theorem [23]. In the proof, we will essentially be considering an n-set partition A = A 1 , .…”

On Erdős–Ginzburg–Ziv inverse theorems

“…Finally, we will need the following partition analog of CDT, which will be our main tool for proving Theorem 1.1 [13], [14]. Theorem 1.4.…”