A multiplicative property for zero-sums. II. (English) Zbl 1502.11018
Let \(G = C_n \oplus C_{nm}\) be an abelian group of rank \(2\), and let \(0 \leq k < n\). It is known that any sequence of \(mn+n-1+k\) elements from \(G\) must contain a zero-sum subsequence of size at most \(mn+n-1-k\). This leads to the inverse problem: what is the structure of a sequence of \(mn+n-2+k\) elements without any zero-sum subsequence of size at most \(mn+n-1-k\)? The answer is known for \(k = 0,1\) and for \(k = n-1\), see the references in the paper.
For \(2 \leq k < n-1\), it is conjectured that a sequence satisfying this condition will have the form \(S=g_1^{[n-1]}g_2^{[sn-1]}(g_1+g_2)^{[(m-s)n+k]}\) for some generating pair \(g_1,g_2\) and some \(1 \leq s \leq m\) (see Theorem 2 for the somewhat more precise statement). The main result in this paper is a reduction of this problem to the case of \(G = C_n \oplus C_n\), as stated in Conjecture 1 (which should be read with \(m = 1\)). In turn, the multiplicativity theorem of Part I [the authors, Discrete Math. 345, No. 10, Article ID 112974, 21 p. (2022; Zbl 1502.11017)] reduces the inverse problem for \(G = C_n \oplus C_n\) to the case of \(n\) being a prime.
For \(2 \leq k < n-1\), it is conjectured that a sequence satisfying this condition will have the form \(S=g_1^{[n-1]}g_2^{[sn-1]}(g_1+g_2)^{[(m-s)n+k]}\) for some generating pair \(g_1,g_2\) and some \(1 \leq s \leq m\) (see Theorem 2 for the somewhat more precise statement). The main result in this paper is a reduction of this problem to the case of \(G = C_n \oplus C_n\), as stated in Conjecture 1 (which should be read with \(m = 1\)). In turn, the multiplicativity theorem of Part I [the authors, Discrete Math. 345, No. 10, Article ID 112974, 21 p. (2022; Zbl 1502.11017)] reduces the inverse problem for \(G = C_n \oplus C_n\) to the case of \(n\) being a prime.
Reviewer: Uzi Vishne (Ramat Gan)
MSC:
| 11B13 | Additive bases, including sumsets |
| 11B75 | Other combinatorial number theory |
| 11B30 | Arithmetic combinatorics; higher degree uniformity |
| 11P70 | Inverse problems of additive number theory, including sumsets |
| 05B10 | Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) |
Citations:
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