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Statement

Setting (p. 155). Signs are $\epsilon\in\{-1,1\}^n$, and a sign sum ($\pm$-sum) of $v_1,\ldots,v_n$ is $\sum_{i=1}^n\epsilon_iv_i$; sign sums are counted with multiplicity, by the number of $\epsilon$ giving them. Work in $\mathbb R^d$ with the euclidean norm (the paper says the argument is the same in any real Hilbert space of dimension $d$). Let $\hat e_1,\ldots,\hat e_d$ be orthonormal, let $m_1,\ldots,m_d$ be odd, and let $v_1,\ldots,v_n$, $n=m_1+\cdots+m_d$, consist of $m_j$ copies of $\hat e_j$ for each $j$.

Lemma 1 (p. 155). If $n$ is odd and $\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}$, then $\epsilon_1+\cdots+\epsilon_n$ is odd, so $\lvert\epsilon_1+\cdots+\epsilon_n\rvert\ge1$.

Lemma 2 (p. 155). For $v_1,\ldots,v_n$ as above, every sign sum has norm at least $\sqrt d$.

The paper draws the consequence (p. 156) that for $d>1$ there are arbitrarily large families of unit vectors none of whose sign sums lies in the unit ball centred at the origin, which disproves Erdős's conjecture that at least $C2^n/n$ sign sums have norm at most $1$. It adds (pp. 156--157) that in dimension two the construction in Proposition 3 forces a counterexample to the bound $1$ to have $n$ even, and that the authors do not know whether the bound $1$ holds when $n$ is required to be odd.

Proof pointer

P. 156. The $j$-th coordinate of a sign sum is a sum of $m_j$ signs, odd by Lemma 1, so every coordinate has absolute value at least $1$.

Dependencies

Lemma 1, which the paper calls trivial and states without proof.

Source. Lemmas 1 and 2, p. 155, 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 setting, both lemmas and the short proof were read clause by clause on the print, pp. 155--157. Nothing here is independently reviewed.

Bears on

• Problem 395: with $d=2$, $m_1=1$ and $m_2=n-1$ odd, the lemma gives unit vectors in the plane with every sign sum of norm at least $\sqrt2$, so for every even $n\ge2$ no sign sum has norm at most $1$, and Erdős's radius $1$ fails; the paper then proposes radius $\sqrt d$ in dimension $d$, whose planar case is the problem's radius $\sqrt2$. It says nothing about the count at radius $\sqrt2$, and the paper leaves radius $1$ for odd $n$ open.