The Erdos discrepancy problem

Abstract

We show that for any sequence $f: {\bf N} \to \{-1,+1\}$ taking values in $\{-1,+1\}$, the discrepancy $$ \sup_{n,d \in {\bf N}} \left|\sum_{j=1}^n f(jd)\right| $$ of $f$ is infinite. This answers a question of Erd\H{o}s. In fact the argument also applies to sequences $f$ taking values in the unit sphere of a real or complex Hilbert space. The argument uses three ingredients. The first is a Fourier-analytic reduction, obtained as part of the Polymath5 project on this problem, which reduces the problem to the case when $f$ is replaced by a (stochastic) completely multiplicative function ${\bf g}$. The second is a logarithmically averaged version of the Elliott conjecture, established recently by the author, which effectively reduces to the case when ${\bf g}$ usually pretends to be a modulated Dirichlet character. The final ingredient is (an extension of) a further argument obtained by the Polymath5 project which shows unbounded discrepancy in this case.