Submit progress

on Erdős #177 · discrepancy / arithmetic progressions

The statement

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 $d$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.