SWhat changed
For real coefficients |x_i|≥1, at most binomial(n,floor(n/2)) of the2^n signed sums fall inside any open interval of length2. Erdős obtains the sharp bound through Sperner's antichain theorem, removes a logarithmic loss for complex-coefficient disk concentration up to an absolute constant, and proves the associated bound by the r largest binomial coefficients for longer intervals and families without an(r+1)-element inclusion chain.
Scientific scope
For real coefficients |x_i|≥1, at most binomial(n,floor(n/2)) of the2^n signed sums fall inside any open interval of length2. Erdős obtains the sharp bound through Sperner's antichain theorem, removes a logarithmic loss for complex-coefficient disk concentration up to an absolute constant, and proves the associated bound by the r largest binomial coefficients for longer intervals and families without an(r+1)-element inclusion chain. Signed sums count sign choices, including repeated equal numerical values; in probabilistic form the signs are independent and equally likely. The real theorem uses the interior of a length2interval, not a closed interval including both endpoints; coefficients have the stated lower magnitude. The complex result is order-sharp with an unspecified absolute constant and integer radius, not the sharp real binomial constant. Hilbert/Banach-space sharp versions and the reverse small-ball assertion are labelled conjectures in this original, not proved here or asserted still open today. Later inverse theorems and arbitrary-sign distributions are not covered. Sperner1928, Littlewood–Offord1943 and Menger disjoint paths are inherited; no new award for those theorems. The original last-page vector and reverse-concentration assertions remain conjectures of that paper, not a claim about current open status.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. Signed sums count sign choices, including repeated equal numerical values; in probabilistic form the signs are independent and equally likely. The real theorem uses the interior of a length2interval, not a closed interval including both endpoints; coefficients have the stated lower magnitude. The complex result is order-sharp with an unspecified absolute constant and integer radius, not the sharp real binomial constant. Hilbert/Banach-space sharp versions and the reverse small-ball assertion are labelled conjectures in this original, not proved here or asserted still open today. Later inverse theorems and arbitrary-sign distributions are not covered. Sperner1928, Littlewood–Offord1943 and Menger disjoint paths are inherited; no new award for those theorems. The original last-page vector and reverse-concentration assertions remain conjectures of that paper, not a claim about current open status.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
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