The Erdős–Ginzberg–Ziv theorem with units

Let x1,…,xrx1,…,xr be a sequence of elements of ZnZn, the integers modulo nn. How large must rr be to guarantee the existence of a subsequence xi1,…,xinxi1,…,xin and units α1,…,αnα1,…,αn with α1xi1+⋯+αnxin=0α1xi1+⋯+αnxin=0? Our main aim in this paper is to show that r=n+ar=n+a is large enough, where aa is the sum of the exponents of primes in the prime factorisation of nn. This result, which is best possible, could be viewed as a unit version of the Erdős–Ginzberg–Ziv theorem. This proves a conjecture of Adhikari, Chen, Friedlander, Konyagin and Pappalardi.We also discuss a number of related questions, and make conjectures which would greatly extend a theorem of Gao.