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on Erdős discrepancy problem (AlphaEvolve problem 40)

The statement

The discrepancy of a sign pattern $a_1,\dots,a_N \in \{-1,+1\}$ is the maximum value of $|a_d + a_{2d} + \dots + a_{kd}|$ for homogeneous progressions $d,\dots,kd$ in $\{1,\dots,N\}$ . For any $D \geq 1$ , let $C(D)$ denote the largest $N$ for which there exists a sign pattern $a_1,\dots,a_N$ of discrepancy at most $C$ . Establish upper and lower bounds on $C(D)$ that are as strong as possible.

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.