A weighted generalization of two theorems of Gao

Abstract

Let G be a finite abelian group and let A ⊆ Z be nonempty. Let DA(G) denote the minimal integer such that any sequence over G of length DA(G) must contain a nontrivial subsequence s1 · · · sr such that r P i=1 wisi = 0 for some wi ∈ A. Let EA(G) denote the minimal integer such that any sequence over G of length EA(G) must contain a subsequence of length |G|, s1 · · · s |G| , such that |G| P i=1 wisi = 0 for some wi ∈ A. In this paper, we show thatconfirming a conjecture of Thangadurai and the expectations of Adhikari, et al. The case A = {1} is an older result of Gao, and our result extends much partial work done by Adhikari, Rath, Chen, David, Urroz, Xia, Yuan, Zeng and Thangadurai. Moreover, under a suitable multiplicity restriction, we show that not only can zero be represented in this manner, but an entire nontrivial subgroup, and if this subgroup is not the full group G, we obtain structural information for the sequence generalizing another non-weighted result of Gao. Our full theorem is valid for more general n-sums with n ≥ |G|, in addition to the case n = |G|.2000 Mathematics Subject Classification. 11B75 (20K01).

Weighted Davenport’s constant and the weighted EGZ Theorem

“…, which implies that (5) and (6) hold for L, which completes the proof of the claim and the statement (8).…”

“…We follow the conventions of [6][7][8] for notation concerning sumsets, sequences and (weighted) subsequence sums over an abelian group. We provide self-contained definitions for all relevant concepts and the weighted Davenport's constant in the subsequent notation section.…”

“…(8) Note that the lower bound E Ψ (G) = |G| + D Ψ (G) − 1 can be easily seen by considering an extremal sequence of length D Ψ (G) − 1 that avoids representing zero using weights from Ψ concatenated with a sequence of |G| − 1 zeros. We see that the following corollary is an immediate consequence of Theorem 1.1.…”

“…We follow the conventions of [6,8] for notation concerning sequences over an abelian group. Let [a, b] = {k ∈ Z|a ≤ k ≤ b}.…”

“…Essentially, the proofs of the statements (7) and (8) are due to Grynkiewicz et al (see [8]). However, while there are some slight change, we give the detailed proofs here for completeness.…”