Finding a solution to the Erdős-Ginzburg-Ziv theorem in $O(n\log \log \log n)$ time

The Erdős-Ginzburg-Ziv theorem states that for any sequence of $2n-1$ integers, there exists a subsequence of $n$ elements whose sum is divisible by $n$. In this article, we provide a simple, practical $O(n\log \log n)$ algorithm and a theoretical $O(n\log \log \log n)$ algorithm, both of which improve upon the best previously known $O(n\log n)$ approach. This shows that a specific variant of boolean convolution can be implemented in time faster than the usual $O(n\log n)$ expected from FFT-based methods.