On the density of sequences of integers the sum of no two of which is a square. II. General sequences

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theorem_b: Lagarias, Odlyzko and Shearer's theorem that there is an absolute constant N_0 such that, for all N >= N_0, every set of integers in [1,N] in which no sum of two distinct elements is a perfect square has at most .475N elements, so every infinite such sequence has upper density at most .475.

theorem_c: Lagarias, Odlyzko and Shearer's circle-method asymptotic, for s >= 2 and 0 < eps < 1/(4(s+1)), for the number of distinct-coordinate integer solutions of 2n = z_0^2 - z_1^2 + ... + z_{2s}^2 with (1-eps)M <= z_i <= M, as M^{2s-1} G_s(2n) f(2n/M^2) up to O(M^{2s-1-delta'}).

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Lagarias, J. C. and Odlyzko, A. M. and Shearer, J. B., On the density of sequences of integers the sum of no two of which is a square. II. General sequences. J. Combin. Theory Ser. A 34 (1983), no. 2, 123--139, doi:10.1016/0097-3165(83)90051-1. The copy read for this card is a re-typeset copy of the paper from an author's website, not the publisher's edition, and prints no copyright or license line on pp. 1--2 or 20--21; the author's publication list that links it (https://www-users.cse.umn.edu/~odlyzko/doc/complete.html, read 2026-10-02) states no copyright, license or terms; the term is unstated.

Source: https://www-users.cse.umn.edu/~odlyzko/doc/complete.html. The copy read is numbered pp. 1--21 (text pp. 1--19, references p. 20, abstract p. 21), and the result pages cite that numbering, not the journal's pp. 123--139.

A set of positive integers has Property NS when no sum of two distinct elements is a perfect square (p. 1), and $d(N)$ is the largest proportion of $[1,N]$ such a set can occupy (p. 2, (1.1)). The paper's main result, Theorem B (p. 2), gives an absolute $N_0$ with $d(N)\le .475$ for all $N\ge N_0$, and hence upper density at most $.475$ for every infinite sequence with Property NS (1.3). The paper recalls Massias's Property NS set of density $\frac{11}{32}$ and the authors' earlier Theorem A, that a union of arithmetic progressions with Property NS has density at most $\frac{11}{32}$ (p. 1); it sees no hope of an upper bound near $\frac{11}{32}$ without new ideas and says that sequences of upper density above $\frac{11}{32}$ may well exist (p. 2). The proof (Section 2, pp. 3--11) bounds the independence number of the graph joining $i$ and $j$ when $i+j$ is a square through a linear programming relaxation by odd-cycle constraints, whose dual weights are counted by Theorem C (p. 7), a circle-method asymptotic for representations of $2n$ by the alternating form $z_0^2-z_1^2+\cdots+z_{2s}^2$ in nearly equal distinct variables; Lemma 4.3 (p. 18) bounds the singular series for $s=7$ between $0.9915$ and $1.0085$. The paper also states, without proof, that an adaptation of the method gives upper density at most $\frac12-c_0(k)$ for sequences no two distinct elements of which sum to a perfect $k$-th power (p. 2), and recalls Erdős's question (its reference [3], p. 3) whether upper density below $\frac12$ follows when sums avoid a sequence $n_1<n_2<\cdots$ with $n_{i+1}/n_i\to1$ that is uniformly distributed modulo every $d$.

Read status: claims checked. Theorems B and C, Lemmas 2.1 and 4.3 and the deduction of (2.44) were read clause by clause on the print and the proof of Section 2 followed; the circle-method proof of Theorem C, a sketch in the paper, was read for structure. Nothing is independently reviewed. Result pages: theorem_b and theorem_c.

Bears on. #438: Theorem B (p. 2) bounds by $.475N$, for $N\ge N_0$, the largest subset of $[1,N]$ in which no sum of two distinct elements is a square, a condition every set whose sumset contains no square meets. The paper recalls Massias's construction of density $\frac{11}{32}$ and does not determine the extremal density.

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