On equitable zero sums

TL;DR

This paper investigates the distribution of zero-sum subsequences in integer sequences modulo N, proving that sufficiently long sequences with odd N contain subsequences with a large number of zero-sum variants.

Contribution

It establishes a new lower bound on the number of zero-sum subsequences in sequences of length at least 4N when N is odd, advancing understanding of zero-sum combinatorics.

Findings

Sequences of length at least 4N with odd N have subsequences with many zero-sum variants.

The number of zero-sum subsequences can be significantly larger than expected in long sequences.

The result applies specifically when N is odd and the sequence length exceeds 4N.

Abstract

It is well-known that any sequence of at least N integers contains a subsequence whose sum is 0 (mod N). However, there can be very few subsequences with this property (e.g. if the initial sequence is just N 1's, then there is only one subsequence). When the length L of the sequence is much longer, we might expect that there are 2^L/N subsequences with this property (imagine the subsequences have sum-of-terms uniformly distributed modulo N -- the 0 class gets about 2^L/N subsequences); however, it is easy to see that this is actually false. Nonetheless, we are able to prove that if the initial sequence has length at least 4N, and N is odd, then there is a subsequence of length L > N, having at least 2^L/N subsequences that sum to 0 mod N.