Abstract

For a finite abelian group G with exp(G)=n and an integer k≥2, Balachandran and Mazumdar (2019) introduced the extremal function fG(D)(k) which is defined to be min{|A|:0̸≠A⊆[1,n−1] withDA(G)≤k} (and ∞ if there is no such A), where DA(G) denotes the A-weighted Davenport constant of the group G. Denoting fG(D)(k) by f(D)(p,k) when G=Fp (for p prime), it is known (Balachandran and Mazumdar, 2019) that p1/k−1≤f(D)(p,k)≤O(plogp)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(p1/k). It was furthermore conjectured that f(D)(p,k)=Θ(p1/k). In this short paper we prove that f(D)(p,k)≤4k2p1/k for sufficiently large primes p.

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