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Source. Erdős (1945), Theorem 2, printed p. 899 (published scan). The explicit constants below are sufficient choices, not constants optimized in the source.

Statement. Let $N\ge1$, let $r\ge1$ be an integer, and let $x_1,\ldots,x_N\in\mathbb C$ satisfy $|x_i|\ge1$. In any open disk of radius $r$, the number $Q$ of sign assignments with sum in the disk satisfies

$$Q\le8rB_N.$$

In particular, the source's strict form holds with $c=9,c_1=18$:

$$Q<9rB_N<18r\,\frac{2^N}{\sqrt N}.$$

The order $2^N/\sqrt N$ cannot be improved uniformly in the inputs, already for fixed radius $r=1$.

Proof. For each $i$, at least one of $|\operatorname{Re}x_i|,|\operatorname{Im}x_i|$ is at least $1/2$: otherwise $|x_i|^2<1/2$, contrary to the hypothesis. Hence one of these coordinate choices works for a set of $t\ge\lceil N/2\rceil$ indices. A common rotation by a right angle if needed makes it the real coordinate. Rotate the target disk by the same amount. Individual sign changes of the selected inputs are absorbed by a bijection of their sign assignments, so, after reindexing, assume

$$\operatorname{Re}x_i\ge\frac12 \qquad(1\le i\le t).$$

Fix the other $N-t$ signs. The first $t$ terms must then have their sum in a translated open disk of radius $r$. Their real parts must lie in an open real interval of length $2r$. Put $y_i=2\operatorname{Re}x_i\ge1$. The corresponding sums $\sum_{i=1}^t\varepsilon_i y_i$ lie in an open interval of length $4r$. The corrected corollary with integer parameter $2r$ bounds their number by $2rB_t$.

There are $2^{N-t}$ choices for the fixed signs. By the elementary binomial estimates,

$$\begin{aligned} Q &\le2r\,2^{N-t}B_t\\ &\le2\sqrt2\,r\,\frac{2^N}{\sqrt t}\\ &\le4r\,\frac{2^N}{\sqrt N}\\ &\le8rB_N. \end{aligned}$$

Since $rB_N>0$ and $B_N\le\sqrt2\,2^N/\sqrt N<2\,2^N/\sqrt N$, the strict displayed inequalities follow.

Finally take even $N$ and $x_1=\cdots=x_N=1$. The unit disk centered at zero contains precisely the assignments with sum zero, of which there are $B_N$. The lower binomial estimate gives $B_N\ge2^N/(2\sqrt N)$, proving the claimed sharp order. $\square$

Scope and precision. The doubling of the selected real coordinates makes the source's application of the real-input corollary explicit. The proof does not use that corollary's incorrect strict endpoint. Positive integer radius is retained. For real radius at least one, rounding upward gives the same order with adjusted constants. A bound proportional to every arbitrarily small positive real radius is impossible: a disk centered on an attainable sum contains that assignment no matter how small the radius is.

The result gives an order bound for complex inputs, not the exact Hilbert-space bound stated as a conjecture in 1945.

Bears on. Problem 498: gives the order $2^N/\sqrt N$ for complex inputs, not the exact bound $B_N$ the problem asks for.