Erdős discrepancy problem

For every infinite sequence $f(1), f(2), f(3), \dots$ taking values in $\{-1, +1\}$ and every constant $C > 0$, there exist positive integers $n$ and $d$ such that $\left|\sum_{j=1}^{n} f(jd)\right| > C$; equivalently, $\sup_{n, d \in \mathbb{N}} \left|\sum_{j=1}^{n} f(jd)\right| = \infty$.

Terence Tao proved the conjecture in September 2015 (published in Discrete Analysis in 2016) by combining the Fourier-analytic reduction from the 2010 Polymath5 collaborative project—which reduced the problem to completely multiplicative functions—with a new logarithmically averaged form of the Elliott conjecture on correlations of multiplicative functions, building on the breakthrough of Kaisa Matomäki and Maksym Radziwiłł.