How often can random signs cancel?
Take several real numbers, each with magnitude at least one, and independently choose a plus or minus sign for each. Add the signed numbers. Sometimes many different choices give the same answer. Anti-concentration asks how large a fraction of all choices can land in a short interval. It is about limiting a pile-up of outcomes, rather than showing most outcomes stay near their average.
Count choices, not distinct answers
Erdős sharp real bound says that among the two-to-the-n sign choices, at most the central binomial coefficient can land inside any open interval of length two. A binomial coefficient counts ways to select a fixed number of objects. Repeated equal numerical sums are still separate sign choices. In probability language, divide the count by two-to-the-n. The lower bound on coefficient magnitudes and the open endpoints are essential parts of this statement.
Four ones make the bound visible
For a teaching example, use four coefficients all equal to one. There are sixteen sign choices. Six choices have two plus signs and two minus signs, producing zero. The open interval from minus one to one contains exactly those six outcomes, and the central binomial coefficient is six. Now try the closed interval from zero to two: it includes six zero outcomes and four outcomes equal to two, giving ten. Changing the endpoint convention would make the claimed bound false.
A probability problem becomes a set problem
Represent each sign choice by the set of positions carrying a plus sign. After making coefficients positive by relabelling signs, a set strictly containing another changes the signed sum by at least two. Two choices inside one open length-two interval therefore cannot have that containment relation. They form an antichain: a collection in which no member contains another. Sperner earlier antichain theorem supplies the central binomial bound. Erdős uses that inherited theorem rather than claiming to invent it.
Check a boundary rather than overclaim
Ask your agent to enumerate the sixteen choices and label which intervals include each answer. Then change one coefficient and repeat, retaining the exact interval endpoints. A finite enumeration checks that example, not every number of coefficients. The original also gives longer-interval and complex-number bounds, but the complex constant is not the same sharp real constant. Its final vector and reverse-concentration proposals were conjectures in that paper; this entry does not declare their present-day status.
Scientific scope
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.
Read the original work
Sources and reading locations for this explanation
- On a lemma of Littlewood and Offord (original1945)
Original1945 pp898–902; Theorem1 and Sperner argument p898, longer intervals and complex result p899, chain-family proof and historical conjectures pp900–902.
This English explanation is an editorial reading view. It does not replace the complete original work or certify its science.
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P. Erdős
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