论公平零和
On equitable zero sums
摘要：
众所周知，任何至少N个整数的序列都包含一个和为0（mod N）的子序列。然而，具有这种性质的序列可能非常少（例如，如果初始序列只有N 1，那么只有一个子序列）。当序列的长度L长得多时，我们可能会期望有2^L/N的连续性具有此性质（假设连续性具有均匀分布的模N项和-0类得到约2^L/N的连续性）;然而，很容易看出这实际上是错误的。尽管如此，我们能够证明，如果初始序列的长度至少为4 N，并且N是奇数，则存在长度为L > N的子序列，其具有至少2^L/N的连续性，其总和为0 mod N。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
