Problem Statement
If $f:\mathbb{N}\to \{-1,+1\}$ then is it true that for every $C>0$ there exist $d,m\geq 1$ such that\[\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert > C?\]
Categories:
Discrepancy
Progress
The Erdős discrepancy problem. This is true, and was proved by Tao [Ta16], who also proved the more general case when $f$ takes values on the unit sphere.In several places (e.g. [Er64b], [Er65b], and [Er81]) Erdős further conjectured that\[\max_{md\leq x}\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert \gg \log x.\]In [Er85c] Erdős also asks about the special case when $f$ is multiplicative.
Source: erdosproblems.com/67 | Last verified: January 13, 2026