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Source. Erdős (1945), printed pp. 898–899 and 901–902 (published scan). All statements on this page are attributed to that paper and date. This page does not undertake a current-status or formalization review.

Hilbert-space concentration, pp. 898–899. For $N$ vectors $x_1,\ldots,x_N$ in Hilbert space with $\|x_i\|\ge1$, Erdős conjectures that the number of sign assignments whose sum lies in any open ball of radius one is at most

$$\binom N{\lfloor N/2\rfloor}.$$

The possible Banach-space extension is a parenthetical suggestion. The statement that he could not even prove an $o(2^N)$ bound for Hilbert-space inputs reports the state of knowledge in 1945. It is not a present-day assertion.

Half-boundary weight, pp. 901–902. For inputs with $|x_i|\ge1$ (the print names no space; the word circle, used for the complex Theorem 2, suggests complex inputs), the proposed strengthening counts assignments in the interior of a unit circle with weight one and assignments on its circumference with weight one half. The proposed bound is again $\binom N{\lfloor N/2\rfloor}$.

This records the literal planar circle formulation. The paragraph uses circles and the modulus $|x_i|$; it does not supply a separate exact Hilbert-ball formulation. The full real special case is proved in this compilation. No proof for general complex inputs is asserted by that page.

Origin-centered lower bound, p. 902, conjecture (1). For $x_1,\ldots,x_N$ with $|x_i|=1$ (the print names no space; this page reads them as complex numbers, as the single-bar modulus and the circle of the preceding half-boundary form suggest), the paper asks for an absolute constant $c>0$ such that

$$\#\left\{\varepsilon\in\{-1,1\}^N: \left|\sum_{i=1}^N\varepsilon_i x_i\right|\le1\right\} >c\,\frac{2^N}{N}.$$

The origin, closed unit disk and exact unit-modulus hypothesis are part of this statement. It is not an assertion for an arbitrary center or for unrestricted inputs of modulus at least one.

Proof scope. These are historical statement pointers. No unproved conjecture is used as an input to Theorems 1–5. The real sharp theorem and complex order bound are proved separately.

Bears on. Problem 498: the problem is the planar case of the Hilbert-space conjecture. Its problem page records the later literature separately. Problem 395 for conjecture (1), which it poses with radius $\sqrt2$ in place of one; its page records the later counterexamples at radius one for even $N$.