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Statement

Setting (first text page, unnumbered): $0<a_1<a_2<\cdots$ is an infinite sequence of integers; $f(n)$ is the number of solutions of $a_i+a_j=n$ and

$$r(n)=f(0)+f(1)+\cdots+f(n),$$

the number of solutions of $a_i+a_j\le n$. The paper introduces the theorem as proving the Erdős–Turán conjecture that $r(n)-cn=O(1)$ cannot hold.

Theorem 1 (first text page, quoted from the 1954 report). "If $c>0$, then

$$r(n)=cn+o\bigl(n^{1/4}\log^{-\frac12-\varepsilon}n\bigr)\qquad(\varepsilon>0) \qquad(1)$$

cannot hold."

So for every $c>0$ and every $\varepsilon>0$ the relation (1) fails. The paper adds on the same page that $c\ne0$ is needed, since with $c=0$ the relation (1) holds for a sequence growing fast enough.

Counting conventions (p. -2-). The paper reads $f(n)$ in three ways: (I) ordered pairs, $i\ne j$ counted twice and $i=j$ once; (II) $i\ne j$ counted once and $i=j$ once; (III) $i\ne j$ counted once and $i=j$ excluded. It states that the theorem holds in each of the three.

Real sequences (first text page). Theorem 1 remains true for sequences of non-negative numbers $\{a_k\}$, not necessarily integers: with $a_k^*$ the nearest integer to $a_k$ and $r^*$ its count, the paper notes $r(n-2)\le r^*(n)\le r(n+2)$ and $r^*(n-2)\le r(n)\le r^*(n+2)$, which transfer the theorem from $r^*$ to $r$.

Companion bound (first text page). Introduced by "Instead of Theorem 1 we can also prove", the page prints

$$\sup_{1\le\ell\le n}\lvert r(\ell)-cn\rvert>\frac{K_\beta n^{1/4}}{\log^\beta n} \qquad\Bigl(\beta>\frac74\Bigr),$$

with $cn$, not $c\ell$, inside the absolute value as printed. By the convention stated at the foot of the page, $K$ is a positive number that may depend on the sequence $\{a_k\}$ but on nothing else; the page does not explain the subscript $\beta$. No proof of this bound is given in the report.

Edition. This page records the 1954 technical-report printing. The journal version's Theorem 1, as the zbMATH review states it, has the larger error term $o(n^{1/4}(\log n)^{-1/2})$; the source card records the difference.

Source. P. Erdős and W. H. J. Fuchs, On a problem of additive number theory, J. London Math. Soc. 31 (1956), 67--73, doi:10.1112/jlms/s1-31.1.67, read in the August 1954 Cornell University technical report printing (Report No. 11, OSR-TN-54-216) identified on the source card: the setting, Theorem 1, the real-sequence remark and the companion bound on the first text page (PDF p. 5), the counting conventions on p. -2- (PDF p. 7), the proof on pp. -4- to -8- (PDF pp. 11--19), read on the page images.

Read depth. Claims checked: the setting, the statement, the counting conventions and the remarks were read clause by clause on the page images. The proof was read on the page images but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. -2- to -8-. Since $r(a_k)$ and $r(2a_k)$ are at least $k(k+1)/2$, relation (1) forces $Ak^2\le a_k\le Bk^2$ for large $k$ (the paper's (2)), and the proof assumes this. The proof is written out only for convention (I) (p. -4-). With $g(z)=\sum_kz^{a_k}$ the generating function of $r$ is $(1-z)^{-1}g(z)^2$, and (1) writes it as $cz(1-z)^{-2}+h(z)$ with the coefficients of $h$ of the size of the error term. On the circle $z=re^{i\theta}$, the triangle inequality, with Schwarz's inequality and Parseval's formula for $h$, bounds the integral of $\lvert(1-z)^{-1}g(z)^2\rvert$ over an arc $\gamma\le\theta\le2\gamma$ from above. From below, the paper's Lemma (p. -3-: for a power series with non-negative coefficients, the mean of $\lvert\phi\rvert^2$ over $\lvert\theta\rvert\le\alpha$ is at least one third of its mean over the whole circle) puts a positive share of the mean of $\lvert g\rvert^2$ near $\theta=0$, while the part with $\theta\le(1-r)^{1/2}$ is only $O(\log\frac1{1-r})$; a dyadic split then finds one arc where the lower bound exceeds the upper one as $r\to1$.

Bears on

• Problem 763: the problem asks whether some $A\subseteq\mathbb N$ and $c>0$ have $\sum_{n\le N}1_A*1_A(n)=cN+O(1)$. That sum is $r(N)$ in convention (I); a bounded error is $o(N^{1/4}\log^{-1/2-\varepsilon}N)$, so Theorem 1 rules it out for infinite $A$ of positive integers, and the real-sequence remark covers an $A$ that contains $0$. For finite $A$ the sum is eventually constant. Theorem 1 thus answers the question no; the problem's claim page credits the theorem in the journal's form.