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Statement

Sign sums ($\pm$-sums) are counted with multiplicity, as on the Lemma 2 page.

Conjecture 4 (p. 156, quoted). "For each integer $d\geq1$, there is a constant $C_d>0$ such that, if $H$ is a real Hilbert space of dimension $d$ and $v_1,\ldots,v_n$ are unit vectors in $H$, then at least $C_d\frac{2^n}{n^{d/2}}$ of their $\pm$-sums have norm at most $\sqrt d$."

The paper proposes it as the adjustment of Erdős's conjecture forced by Lemma 2 (radius $1$ fails for $d>1$) and Proposition 3 (the rate $2^n/n$ fails for $d>2$). It notes (p. 156) that $d=2$ is the only case keeping the rate $\Omega(2^n/n)$ and that this case already seems hard with the radius $\sqrt d$. The paper proves no case of the conjecture; its Proposition 8 is a weak version with the ball not centred at the origin.

Source. Conjecture 4, p. 156, of W. Carnielli and P. K. Carolino, Adjusting a conjecture of Erdős, Contrib. Discrete Math. 6 (2011), no. 1, 154--159, as identified on the source card.

Read depth. Claims checked: the conjecture and the surrounding remarks were read clause by clause on the print, pp. 156--157. Nothing here is independently reviewed.

Bears on

• Problem 395: the case $d=2$ of the conjecture, with $\mathbb C$ read as the real plane, is the problem's statement: at least $C_22^n/n$ sign choices give $\lvert\epsilon_1z_1+\cdots+\epsilon_nz_n\rvert\le\sqrt2$. The paper poses it and leaves it open.