On the Erdos Discrepancy Problem

TL;DR

This paper advances the understanding of the Erdős discrepancy problem by proving that all sufficiently long completely multiplicative sequences have discrepancy at least 4, confirming the conjecture for discrepancies up to 3, and improving bounds for sequences with higher discrepancies.

Contribution

It proves the Erdős discrepancy conjecture for completely multiplicative sequences with discrepancy up to 3 and significantly extends the known maximum length of such sequences.

Findings

Sequences of length ≥127,646 have discrepancy ≥4

Longest sequence with discrepancy 3 is now at least 127,645 in length

Provides methods to construct sequences with higher discrepancies

Abstract

According to the Erd\H{o}s discrepancy conjecture, for any infinite $\pm 1$ sequence, there exists a homogeneous arithmetic progression of unbounded discrepancy. In other words, for any $\pm 1$ sequence $(x_1,x_2,...)$ and a discrepancy $C$, there exist integers $m$ and $d$ such that $|\sum_{i=1}^m x_{i \cdot d}| > C$. This is an $80$-year-old open problem and recent development proved that this conjecture is true for discrepancies up to $2$. Paul Erd\H{o}s also conjectured that this property of unbounded discrepancy even holds for the restricted case of completely multiplicative sequences (CMSs), namely sequences $(x_1,x_2,...)$ where $x_{a \cdot b} = x_{a} \cdot x_{b}$ for any $a,b \geq 1$. The longest CMS with discrepancy $2$ has been proven to be of size $246$. In this paper, we prove that any completely multiplicative sequence of size $127,646$ or more has discrepancy at least $4$, proving the Erd\H{o}s discrepancy conjecture for CMSs of discrepancies up to $3$. In addition, we prove that this bound is tight and increases the size of the longest known sequence of discrepancy $3$ from $17,000$ to $127,645$. Finally, we provide inductive construction rules as well as streamlining methods to improve the lower bounds for sequences of higher discrepancies.