A new exponential upper bound for the ErdH{o}s-Ginzburg-Ziv constant

Naslund used Tao's slice rank bounding method to give new exponential upper bounds for the Erd\H{o}s--Ginzburg-Ziv constant of finite Abelian groups of high rank. In our short manuscript we improve slightly Naslund's upper bounds. We extend Naslund's results and prove new exponential upper bounds for the Erd\H{o}s--Ginzburg-Ziv constant of arbitrary finite Abelian groups. Our main results depend on a conjecture about Property D.