Sequences in abelian groups G of odd order without zero-sum subsequences of length exp(G)

Abstract

We present a new construction for sequences in the finite abelian group C r n without zero-sum subsequences of length n, for odd n. This construction improves the maximal known cardinality of such sequences for r > 4 and leads to simpler examples for r > 2. Moreover we explore a link to ternary affine caps and prove that the size of the second largest complete caps in AG(5, 3) is 42.

On the Erdős–Ginzburg–Ziv constant of finite abelian groups of high rank

“…Then (s(G) − 1)/2 equals the maximal size of a cap in the affine space F r 3 . The maximal size of such caps has been studied in finite geometry for decades, and the precise value is known so far only for r 6 (see [18,2]). The connection to affine caps will be addressed in greater detail in Section 4.…”

On zero sum subsequences of restricted size

“…We got to know from Dr Wolfgang Schmid that there doesn't seem to be a general process available to get the lower bound on s(C r nm ) in this situation. For example note, s(C 5 3 ) = 45(3 − 1) + 1 is known, but he thinks nobody knows s(C 5 9 ) and there are reasons to believe that they are not related in the form one might expect; for s(C 5 3 ) = 45(3 − 1) + 1 is known but the best lower bound for s(C 5 9 ) is 42(9 − 1) + 1 and the upper bound is 45(9 − 1) + 1. One can see the paper by Edel (Theorem 1 of [5]) for information on this.…”

“…For example note, s(C 5 3 ) = 45(3 − 1) + 1 is known, but he thinks nobody knows s(C 5 9 ) and there are reasons to believe that they are not related in the form one might expect; for s(C 5 3 ) = 45(3 − 1) + 1 is known but the best lower bound for s(C 5 9 ) is 42(9 − 1) + 1 and the upper bound is 45(9 − 1) + 1. One can see the paper by Edel (Theorem 1 of [5]) for information on this. Also it has been proved that, s(C 3 3 ) = 9(3) − 8 = 19 (see Satz 4 of [14]); also see Corollary 4.5 of [6].…”

Inverse zero-sum problems

“…For r = 1 we have c(G) = 2 and for r = 2 we have c(G) = 4 (see Theorem 2.4). In the case of higher ranks bounds for c(G) were given by N. Alon and M. Dubiner (see [1]) and then by Y. Edel, C. Elsholtz, A. Geroldinger, S. Kubertin and L. Rackham (see [32,15,13,12]). We make use of the simple fact that η(C r 2 ) = 2 r and s(C r 2 ) = 2 r + 1 (see [30,Corollary 5.7.6]).…”