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Source. Erdős (1945), printed pp. 898–902 (published scan). The notation below separates the source's ground-set size from its occasionally inconsistent central-rank variable.

For $N\ge1$, inputs $x_1,\ldots,x_N$ and an assignment $\varepsilon\in\{-1,1\}^N$, write

$$Z(\varepsilon)=\sum_{i=1}^N\varepsilon_i x_i, \qquad B_N=\binom N{\lfloor N/2\rfloor}.$$

Every count is a count of assignments. If distinct assignments give the same value of $Z$, each is counted. The inputs may be repeated.

An open interval of length $2r$ is $(u-r,u+r)$. An open disk of radius $r$ is $\{z:|z-w|<r\}$. The real and complex concentration theorems use positive integer $r$; their statements do not include the endpoints. Half-open intervals of length two also satisfy Theorem 1, as proved there.

For the combinatorial results, $[N]=\{1,\ldots,N\}$, with $[0]=\varnothing$. A family is a set of distinct subsets of $[N]$, and the rank of a member $A$ is $|A|$. A chain uses strict inclusions. Indexed repetitions of the same subset are not allowed in these family bounds.

For an integer $r\ge0$, let $S(N,r)$ be the sum of the largest $\min(r,N+1)$ coefficients of $(1+x)^N$, with $S(N,0)=0$. Coefficients at different ranks remain separate even when their values are equal. For $1\le r\le N+1$, put

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

Then

$$S(N,r)=\sum_{k=L}^U\binom Nk.$$

Indeed, the coefficients are symmetric and $\binom N{k+1}/\binom Nk=(N-k)/(k+1)$, so these are $r$ consecutive largest ranks. In a tied case the adjacent central choice has the same sum. For $r\ge N+1$, the convention gives $S(N,r)=2^N$. This truncation and the empty-parameter cases are explicit elementary extensions of the source's phrase “the $r$ largest binomial coefficients” (Theorems 4 and 5, pp. 899–900).

Bears on. Problem 498.