Kleitman: the Littlewood–Offord plane bound

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higher_dimensional_scope: Records the orthant argument and Lemmas III–IV as source claims and pointers, with their uncompiled geometric scope.

lemma_i: Partitions the Boolean lattice into saturated symmetric chains, with the empty ground set and empty residual chain handled explicitly.

lemma_ii: Bounds a union of q antichains by the q largest binomial levels, including equality and the zero or oversized-q cases.

notation: Defines symmetric chains, binomial tails and signed-sum multiplicity, and separates the plane proof from the uncompiled geometric branch.

open_disc_transfer: Transfers the strict-norm theorem to open discs by finite scaling and records why the corresponding closed-disc statement is false.

remark_p253: Counts chains with at least a prescribed number of members by a single rank level, with exact parity and endpoint conventions.

theorem_i: Proves the sharp middle-binomial bound for sign choices in a closed unit disc when all complex coefficients have modulus greater than one.

theorem_ii: Bounds families excluding comparable pairs that differ in just one color class, using symmetric chains and an exact middle-level count.

theorem_iii: States Kleitman's claim that in each finite-dimensional real space the signed sums of n vectors of length greater than one in a unit ball number at most the middle binomial coefficient once n is large, with the printed strict inequality refuted and the proof left unchecked.

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Daniel J. Kleitman, On a lemma of Littlewood and Offord on the distribution of certain sums, Mathematische Zeitschrift 90 (1965), no. 4, 251–259, DOI 10.1007/BF01158565; the issue number and DOI are from the Crossref record. Received 19 November 1964.

Source and version

The copy read for this card is the GDZ scan of the published article, with its archive cover preserved. It has ten physical pages: cover, then printed 251–259 on PDF pages 2–10. The source record identifies the primary download and IIIF provenance. The prepended archive cover prints the digitizer's terms, "The Goettingen State and University Library provides access to digitized documents strictly for noncommercial educational, research and private purposes" and "Publication and/or broadcast in any form (including electronic) requires prior written permission from the Goettingen State- and University Library", while the article pages print no copyright line, every other right reserved.

The GDZ article download preserves the original journal pages; the EuDML record corroborates author, title, volume and pagination. The article is in English despite the archive's German language label. File-generation and digitization dates are not alternate mathematical versions. No other version of this 1965 paper is compared here.

Results and method

Theorem I proves that, for complex $a_i$ of modulus strictly greater than one, at most $\binom n{\lfloor n/2\rfloor}$ sign choices have their sum in a closed unit disc. Sign choices are counted with multiplicity even when the sums coincide. Equal real coefficients show sharpness. The finite scaling consequence gives the same bound for moduli at least one in an open unit disc. The closed norm-one variant fails already at $n=1$.

The proof is a subset-family argument. Sign reindexing into the upper half-plane and a split by quadrant divide the coefficients into two classes with nonnegative pairwise inner products within each class; the compiled proof adds a generic rotation, which the source does not use, so that no coefficient lies on an axis. Two comparable sign-index sets differing in only one class would produce sums more than two apart. The two-color Sperner theorem bounds families with this exclusion.

Its proof uses Lemma I's symmetric-chain decomposition, the exact chain-length counts, and Lemma II's bound for a union of antichains. The source's proof ends with a binomial convolution identity, which the compiled proof derives by counting the middle level of products of symmetric chains, with all parity cases included.

Proof coverage and source precision

Six components have complete rewritten proofs here: Lemma I, the chain-count remark, Lemma II, Theorem II, Theorem I, and the open-disc transfer. The shared notation specifies the empty ground set, chain lengths measured in members, zero or oversized numbers of antichains, and sign multiplicity. Empty residual chains are discarded. The chain-tail threshold is “at least,” and all family bounds are non-strict; the source's contrary prose is identified on the relevant pages.

These are explicit compilation expansions and source-precision corrections, not author-issued errata. The full plane chain is self-contained relative to elementary finite counting and Euclidean geometry. Although the source attributes Lemma II to Erdős 1945, its proof is included locally.

The higher-dimensional material on pp. 254–259 remains at statement or qualified source-claim and proof-pointer scope. In particular, no complete geometric proof of Lemma IV or Theorem III, the eventual bound in each finite dimension, is claimed. Theorem III's literal strict count conflicts with its equality examples and weak proof conclusion; correcting that wording does not close the remaining geometric arguments. The separate 1970 paper and later refinements are not compiled as part of this 1965 source.

Bears on

• Problem 498: Theorem I bounds by $\binom n{\lfloor n/2\rfloor}$ the sign choices whose sum lies in a unit disc when every $|a_i|>1$; the open-disc transfer, a deduction of this card rather than a statement of the paper, gives the problem's open-disc reading with $|z_i|\ge1$. Theorem III gives the same bound in each finite dimension only for large $n$ and adds nothing to the plane case.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.