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Statement

Theorem 2 (p. 117). Let $N>N_1$ and let $\mathcal A\subset\{1,2,\ldots,N\}$ be such that $a+a'$ is squarefree for all $a\in\mathcal A$, $a'\in\mathcal A$. Then $|\mathcal A|<3N^{3/4}\log N$ (display (2)).

The print's display (2) reads $\mathcal A<3N^{3/4}\log N$, without the cardinality bars; the proof (p. 122) bounds the number of elements. The threshold $N_1$ is not made explicit.

Proof pointer

Sections 3 and 4, pp. 120--122. Lemma 1 (p. 120) is the large sieve inequality, taken from Montgomery's Topics in Multiplicative Number Theory (Corollary 2.2, p. 12). Lemma 2 (p. 120) is a large sieve by squares of primes derived from it: for integers $M$ and $N\ge1$ and a set of $Z$ integers in $[M+1,M+N]$, with $Z(q,h)$ the number of its elements congruent to $h$ modulo $q$, $\sum_{p^2\le Q}p^2\sum_{h=1}^{p^2}\bigl(Z(p^2,h)-Z/p^2\bigr)^2\le(Q^2+\pi N)Z$ for every $Q>0$. For Theorem 2 (section 4), $a+a'\not\equiv0\pmod{p^2}$ confines $\mathcal A$ to at most $(p^2-1)/2$ residue classes modulo $p^2$, so at least $(p^2+1)/2$ classes are empty and the left side of Lemma 2 is at least $\frac{Z^2}{2}\pi(Q^{1/2})$. Taking $Q=N^{1/2}$ and the prime number theorem give $Z<3N^{3/4}\log N$ for large $N$.

Read depth

Claims checked: the statement and Lemmas 1 and 2 were read clause by clause on the page images of the print, and the derivation of Theorem 2 from Lemma 2 on pp. 121--122 was followed. The proof of Lemma 2 was read for structure; Lemma 1 is cited, not proved, in the paper. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs: the large sieve inequality and the identities for $S(a/q)$ that the paper takes from Montgomery's book (pp. 12, 23 and 24 there), and the prime number theorem.

Source. P. Erdős and A. Sárközy, On divisibility properties of integers of the form $a+a'$, Acta Math. Hungar. 50 (1987), no. 1--2, 117--122, doi:10.1007/BF01903370; the edition read is named on the source card.

Bears on

• Problem 1109: gives $f(N)<3N^{3/4}\log N$ for $N>N_1$. This is far from the bounds $N^{o(1)}$ and $(\log N)^{O(1)}$ the problem asks about, so it answers neither question.
• Problem 1103: the paper says nothing about infinite sequences. Applied to the terms up to $N$ of an infinite sequence of positive integers whose pairwise sums are all squarefree, Theorem 2 bounds their number by $3N^{3/4}\log N$ for $N>N_1$, which forces growth $a_j\ge j^{4/3-o(1)}$; this does not settle how fast such a sequence must grow.