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Claim. For every $d\ge2$ there is a constant $c_d>0$ such that for all unit vectors $v_1,\ldots,v_n\in\mathbb R^d$ and independent uniform signs $\epsilon_i\in\{-1,1\}$,

$$\Pr\Bigl[\Bigl\lVert\sum_{i=1}^n\epsilon_iv_i\Bigr\rVert\le\sqrt d\Bigr]\ge c_dn^{-d/2}.$$

The case $d=2$ is the question of [[problems/analysis/E0395/_index|Problem 395]]: the closed disk of radius $\sqrt2$ and a bound of order $1/n$. The theorem is J. Beck, On a geometric problem of Erdős, Sárközy, and Szemerédi concerning vector sums, European J. Combin. 4 (1983), no. 1, 1–10, proved by harmonic analysis. The paper is not held here and has not been read; the statement is recorded as two independent later papers quote it, and the two quotations differ. Theorem 1.2 on p. 2 of He, Juškevičius, Narayanan and Spiro gives it for $d\ge2$ and unit vectors, the form recorded above; Theorem 1.2 on p. 2 of Hollom, Portier and Souza gives it for every $d\ge1$ and vectors of norm at most $1$, a stronger form that contains the recorded one.

Acceptance. The paper is a refereed journal publication, the refereed evidence. The site's curator credits the solution of the problem to He, Juškevičius, Narayanan and Spiro, whose claim page carries that credit, and does not name Beck; no reviewer independent of the author is recorded for this paper, so no reviewed evidence is listed. The publisher's record dates the issue to March 1983 and gives no day; this page is dated to the first day of that month.

Depends on. Nothing beyond the cited paper.