Reflections on the Erd\H {o}s Discrepancy Problem

TL;DR

This paper explores coloring problems related to the Erdős Discrepancy Problem, proving balanced colorings for certain sets, analyzing restricted variants, and connecting to deep questions about multiplicative functions with bounded partial sums.

Contribution

It establishes the existence of 2-colorings balancing homogeneous arithmetic progressions for fixed lengths and discusses variants and related deep number-theoretic questions.

Findings

Existence of 2-colorings balancing sets for fixed k

Analysis of restricted coloring variants

Connections to multiplicative functions with bounded partial sums

Abstract

We consider some coloring issues related to the famous Erd\H {o}s Discrepancy Problem. A set of the form $A_{s,k}=\{s,2s,\dots,ks\}$, with $s,k\in \mathbb{N}$, is called a \emph{homogeneous arithmetic progression}. We prove that for every fixed $k$ there exists a $2$-coloring of $\mathbb N$ such that every set $A_{s,k}$ is \emph{perfectly balanced} (the numbers of red and blue elements in the set $A_{s,k}$ differ by at most one). This prompts reflection on various restricted versions of Erd\H {o}s' problem, obtained by imposing diverse confinements on parameters $s,k$. In a slightly different direction, we discuss a \emph{majority} variant of the problem, in which each set $A_{s,k}$ should have an excess of elements colored differently than the first element in the set. This problem leads, unexpectedly, to some deep questions concerning completely multiplicative functions with values in $\{+1,-1\}$. In particular, whether there is such a function with partial sums bounded from above.