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Claim. The manuscript Polynomial bounds for infinite-dimensional vector balancing (dated 18 September 2026, ten pages) states as its Theorem 1 that for every sequence $v_1,v_2,\ldots$ of real vectors with countably many coordinates and $\lvert v_n(k)\rvert\le1$ there are signs $\delta_n\in\{-1,1\}$ with

$$\sup_{m\ge1}\left\lvert\sum_{n=1}^{m}\delta_n v_n(k)\right\rvert \le C\,k^{3/2+\sqrt2}\qquad(k\ge1)$$

for an absolute constant $C$. Its Corollary 2 applies this to the membership vectors $v_n(i)=\mathbf{1}_{n\in A_i}$ of the sets of Problem 178: there is one $f:\mathbb{N}\to\{-1,1\}$ whose partial sums along each of the first $d$ sequences are at most $C d^{3/2+\sqrt2}$, improving the exponent $4+\epsilon$ of Beck's 2017 quantitative bound. By the manuscript's introduction and the submission's summary, the method keeps the two outer parts of Beck's proof, the grouping of vectors into signed blocks at a hierarchy of scales and the compactness step, and replaces the finite cancellation inside each block: where Beck cancels the early coordinates by pigeonhole and bounds the later ones by the block size, an entropy partial-coloring step on weighted coordinates signs a constant fraction of $K$ vectors of Hilbert norm at most $B$ so that the sum has norm $O(\sqrt{K}B)$ and each later coordinate stays $O(B)$; carrying the norm estimate and the coordinate estimate through the grouping separately, and choosing the scales to balance them, gives the exponent. The manuscript's second result, a $\pm1$ coloring of the positive integers whose discrepancy on each finite arithmetic progression is a fixed power of its common difference, concerns Problem 177 and is recorded on its own page.

Submission note. Posted to erdosproblems.com as a proof claim by Samuel Korsky (account SamKorsky) on 19 September 2026, giving "GPT Astra" as the AI used:

The paper proves that every sequence $v_n\in[-1,1]^{\mathbb N}$ admits a single signing satisfying

$$\sup_m\left|\sum_{n\le m}\delta_n v_n(k)\right| > =O(k^{3/2+\sqrt2})$$

for every coordinate $k$, improving Beck’s exponent

$4+\varepsilon$. The proof retains Beck’s recursive grouping and compactness argument but strengthens the finite cancellation step. Beck uses pigeonhole cancellation on the early coordinates and bounds the remaining coordinates by group size. Here an entropy partial-coloring argument, after weighting the coordinates, merges a positive proportion of $K$ vectors of Hilbert norm at most $B$, obtaining norm $O(\sqrt K B)$ and individual later-coordinate bounds $O(B)$. Propagating these two bounds separately and optimizing the grouping scales yields the improved exponent. Notes: We also construct a coloring of the integers with discrepancy $O(d^{5/2+2\sqrt2})$ on every finite arithmetic progression of common difference $d$, improving Beck’s $8+\varepsilon$ in Problem #177.

Scope. Full. Corollary 2 is the problem's statement with the explicit bound $\ll d^{3/2+\sqrt2}$ in place of Beck's $\ll d^{4+\epsilon}$; the existence of a bounded $f$, the problem's question, is already the accepted claim Beck 1981, and what is new here is the exponent.

Depends on. Nothing in this wiki; the manuscript's own argument carries the claim.

Standing. Claimed. The result was submitted to the site's proof-claim tab on 19 September 2026, attributed to Samuel Korsky, whose acknowledgment credits the system GPT Astra with the mathematical insights behind the improvements, and links a manuscript dated 18 September 2026 and hosted on a file-sharing service. On 2026-10-07 the site's tab carried no comment on it; the site says that listing a proof claim is no guarantee of correctness and implies no examination by anyone associated with it, and no review of the manuscript is known.