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$500

If $f:\mathbb{N}\to \{-1,+1\}$ then is it true that for every $C>0$ there exist some $d,m\geq 1$ such that
\[\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert > C?\]

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.

Find the smallest $h(d)$ such that the following holds. There exists a function $f:\mathbb{N}\to\{-1,1\}$ such that, for every $d\geq 1$,
\[\max_{P_d}\left\lvert \sum_{n\in P_d}f(n)\right\rvert\leq h(d),\]
where $P_d$ ranges over all finite arithmetic progressions with common difference $d$.

Cantor, Erdős, Schreiber, and Straus [Er66] proved that $h(d)\ll d!$ is possible. Van der Waerden's theorem implies that $h(d)\to \infty$. Beck [Be17] has shown that $h(d) \leq d^{8+\epsilon}$ is possible for every $\epsilon>0$. Roth's famous discrepancy lower bound [Ro64] implies that $h(d)\gg d^{1/2}$.