The Weighted Davenport constant of a group and a related extremal problem II (original) (raw)

2019, arXiv (Cornell University)

For a finite abelian group G with exp(G) = n and an integer k ≥ 2, Balachandran and Mazumdar [3] introduced the extremal function f (D) G (k) which is defined to be min{|A| : ∅ = A ⊆ [1, n − 1] with D A (G) ≤ k} (and ∞ if there is no such A), where D A (G) denotes the A-weighted Davenport constant of the group G. Denoting f (D) G (k) by f (D) (p, k) when G = F p (for p prime), it is known ([3]) that p 1/k − 1 ≤ f (D) (p, k) ≤ O k (p log p) 1/k holds for each k ≥ 2 and p sufficiently large, and that for k = 2, 4, we have the sharper bound f (D) (p, k) ≤ O(p 1/k). It was furthermore conjectured that f (D) (p, k) = Θ(p 1/k). In this short paper we prove that f (D) (p, k) ≤ 4 k 2 p 1/k for sufficiently large primes p.

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