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Claim. For vectors $x_1,\ldots,x_n$ of norm at least one in a Hilbert space, at most $\binom n{\lfloor n/2\rfloor}$ of the $2^n$ sign choices put $\sum_i\epsilon_ix_i$ in any given open ball of radius one. This is the Hilbert-space conjecture of Erdős's 1945 paper, pp. 898–899 (the conjecture as stated). D. J. Kleitman resolved it in On a lemma of Littlewood and Offord on the distributions of linear combinations of vectors, Adv. Math. 5 (1970), no. 1, 155–157. The site credits the paper with the generalization to arbitrary Hilbert spaces. The introductions of He, Juškevičius, Narayanan and Spiro (card, arXiv:2408.11034v3, p. 2) and of Hollom, Portier and Souza (card, arXiv:2503.24202v1, p. 1) state that it resolved the conjecture, and the former add that Kleitman extended the bound to arbitrary normed spaces. The paper itself has not been read for this page; the statement is recorded as those sources give it. In the plane it is Problem 498 with $D$ an open disc and moduli at least one, with no scaling step.

Acceptance. A refereed journal publication, the refereed evidence. The publisher's record dates the issue to August 1970 and gives no day, so the page is dated to the first day of that month. The site's curator, Thomas Bloom, labels the problem proved and credits this paper with the Hilbert-space generalization, the reviewed evidence.

Depends on. Nothing beyond the cited paper.