The Erdős-Ginzburg-Ziv constant 
of a finite Abelian group 
of group exponent 
is the least length such that every sequence
over 
of that length contains a zero-sum subsequence
of exactly 
terms (Gao and Geroldinger 2006). The length is prescribed by the group
exponent, whereas the Gao constant prescribes
the group order. For the cyclic
group 
,
the Erdős-Ginzburg-Ziv theorem
gives 
.
For 
and 
with 
and odd 
,
Wang and Zeng (2026) proved
 |
(1)
|
For 
and 
with 
,
Elsholtz (2026) proved
 |
(2)
|
Elsholtz (2026) strengthened these bounds in the revised treatment. For 
with 
, odd 
, and 
,
 |
(3)
|
where 
is the floor function. This improves the Wang-Zeng
bound for 
.
For 
with odd 
,
Elsholtz (2026) also proved
 |
(4)
|
for 
.
In particular, 
.
Here 
is the direct sum of 
copies of 
. For even 
, these strict improvements on 
obstruct Property D.
Elsholtz (2026) reports AI assistance with the constructions and finite verification.
Han and Zhang (2026) proved the dimension-dependent bound
 |
(5)
|
for all positive integers 
and 
. Thus any sequence of 
lattice points in 
contains 
terms whose centroid
is a lattice point. Repeated terms are allowed and are counted by their positions.
Han and Zhang (2026) report ChatGPT assistance in simplifying proof strategies.
for Even 
:
Rigid Sequences, Products, and Growth Beyond Harborth's Bound." 27 Oct 2026.
." 29 Oct 2026.