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Claim. J. Beck, A discrepancy problem: balancing infinite dimensional vectors, in Number Theory -- Diophantine Problems, Uniform Distribution and Applications (Springer, 2017), 61--82, cited as [Be17] on the problem page. The chapter's abstract states the result as a corollary of a general theorem on balancing infinite-dimensional vectors in the maximum norm: for every $\epsilon>0$ there is a function $g:\mathbb N\to\{-1,1\}$ such that, with $D_g(d)=\max_{a\ge1,m\ge1}\lvert\sum_{i=0}^{m-1}g(a+id)\rvert$ the largest sum along a finite arithmetic progression of common difference $d$, $D_g(d)\le d^{8+\epsilon}$ for all sufficiently large $d\ge d_0(\epsilon)$, independently of the starting point and the length of the progression; in the notation of Problem 177, $h(d)\le d^{8+\epsilon}$ is attainable for every $\epsilon>0$. The abstract adds that Roth's theorem gives $D_g(d)\ge\sqrt d/20$ for every $g$, so a polynomial bound is the right order. The argument enumerates the residue classes modulo each $d$ by increasing modulus, writes a progression as the difference of two prefixes of one class, and balances the membership vectors of the classes, a method Korsky's 2026 manuscript on its claim page describes as the one it refines. The chapter has no library card.

Covers. The upper bound $h(d)\le d^{8+\epsilon}$, for every $\epsilon>0$ and all $d\ge d_0(\epsilon)$, alone. The problem asks for the smallest $h(d)$, and the chapter determines neither its order nor a lower bound; Roth's bound on its claim page leaves the exponent between $1/2$ and $8$, and Korsky's claimed bound lowers the upper exponent to $5/2+2\sqrt2$.

Depends on. No page of this wiki.

Standing. Claimed. The chapter appears in an edited Springer volume, a Festschrift, published online on 30 May 2017; no evidence that the volume was refereed is recorded here, so no refereed evidence is listed. The site's curator credits the bound to [Be17] in the problem's commentary, but the site labels the problem OPEN, so that credit is not reviewed evidence. The proof is not checked by this corpus, and nothing is independently reviewed by this project.