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Source. Erdős (1945), Theorem 3, printed p. 899 (published scan).

Statement. Let $N\ge1$, let $r\ge1$ be an integer, and let $x_1,\ldots,x_N$ be real numbers with $|x_i|\ge1$. The number of sign assignments whose sum lies in any open interval of length $2r$ is at most $S(N,r)$, the sum of the largest $\min(r,N+1)$ coefficients of $(1+x)^N$. The bound is sharp.

Proof. As in Theorem 1, make all inputs positive by reindexing their signs. If $A$ is the set of positive coordinates, its sum is $Z_A=2\sum_{i\in A}x_i-\sum_i x_i$. For $A\subsetneq D$ with $|D\setminus A|\ge r$,

$$Z_D-Z_A=2\sum_{i\in D\setminus A}x_i\ge2r.$$

Such a pair cannot give two sums in the same open interval of length $2r$. The subsets corresponding to qualifying assignments therefore satisfy Theorem 4, which bounds their number by $S(N,r)$.

For sharpness when $1\le r\le N+1$, let all $x_i=1$ and use

$$L=\left\lfloor\frac{N-r+1}{2}\right\rfloor, \qquad U=L+r-1.$$

The open interval

$$(2L-N-1,\;2U-N+1)$$

has length $2r$ and contains precisely the rank sums $2k-N$ with $L\le k\le U$. Their assignment multiplicities add to $\sum_{k=L}^U\binom Nk=S(N,r)$. For $r>N+1$, the interval $(-r,r)$ contains every sum of the all-one inputs, so the truncated value $2^N$ is attained. $\square$

Conventions. The integer-radius and truncation conventions are explicit in the notation page. This is the source's sharper replacement for its coarse interval corollary.

Bears on. Problem 498: at $r=1$ this is Theorem 1, the problem's bound for real inputs.