# Good weights for the Erd\H{o}s discrepancy problem

## TL;DR

This paper extends the Erdős discrepancy problem to weighted variants involving structured and random weights, proving unboundedness of weighted sums of multiplicative functions and employing measure-preserving systems analysis.

## Contribution

It introduces weighted versions of the Erdős discrepancy problem, combining structured and random weights, and develops new structural results for measure-preserving systems related to multiplicative functions.

## Key findings

- Weighted sums of bounded multiplicative functions are unbounded.
- Structured weights with irrationality features lead to unbounded discrepancy.
- The analysis leverages measure-preserving systems associated with multiplicative functions.

## Abstract

The Erd\H{o}s discrepancy problem, now a theorem by T. Tao, asks whether every sequence with values plus or minus one has unbounded discrepancy along all homogeneous arithmetic progressions. We establish weighted variants of this problem, for weights given either by structured sequences that enjoy some irrationality features, or certain random sequences. As an intermediate result, we establish unboundedness of weighted sums of bounded multiplicative functions and products of shifts of such functions. A key ingredient in our analysis for the structured weights, is a structural result for measure preserving systems naturally associated with bounded multiplicative functions that was recently obtained in joint work with B. Host.