A note on high-dimensional discrepancy of subtrees

TL;DR

This paper establishes precise asymptotic bounds for the high-dimensional discrepancy of trees and confirms related conjectures, advancing understanding of imbalance measures in combinatorial structures.

Contribution

It provides tight bounds for the discrepancy of trees and resolves two conjectures on high-dimensional and oriented discrepancy of subtrees.

Findings

Established tight asymptotic bounds for tree discrepancy.

Confirmed conjectures by Krishna et al. on high-dimensional and oriented discrepancy.

Advanced theoretical understanding of subtree imbalance measures.

Abstract

For a tree $T$ and a function $f \colon E(T)\to \mathbb{S}^d$, the imbalance of a subtree $T'\subseteq T$ is given by $|\sum_{e \in E(T')} f(e)|$. The $d$-dimensional discrepancy of the tree $T$ is the minimum, over all functions $f$ as above, of the maximum imbalance of a subtree of $T$. We prove tight asymptotic bounds for the discrepancy of a tree $T$, confirming a conjecture of Krishna, Michaeli, Sarantis, Wang and Wang. We also settle a related conjecture on oriented discrepancy of subtrees by the same authors.