---

Statement

Throughout the paper (printed p. 28) $1\le a_1<a_2<\cdots$ is a sequence of integers, $A(x)=\sum_{a_i\le x}1$, and a sequence has density $0$ when $A(x)/x\to0$. Condition (1) is that the equation

$$a_k=a_{i_1}+a_{i_2}+\cdots+a_{i_r},\qquad i_1<i_2<\cdots<i_r, \tag{1}$$

has no solution for any $k$ and $r$: no term is a sum of distinct other terms (the display is printed with the index condition $i_1<\cdots<i_r$ alone; the English summary on printed p. 38 writes it with $i_r<k$, and Theorem I's hypothesis, that no $a$ splits into a sum of distinct $a$'s, implies the same; the site's Problem 876 writes the same condition as $a=b_1+\cdots+b_r$ with $b_1<\cdots<b_r<a$ in $A$).

Theorem I (printed p. 28). If no $a$ can be written as a sum of distinct other $a$'s, then $A$ has density $0$.

Theorem II (printed p. 30). If (1) has no solution, then $\sum_{i=1}^\infty1/a_i$ converges, and indeed always

$$\sum_{i=1}^\infty\frac1{a_i}<103.$$

After the proof (printed p. 31): it would be easy to improve $103$ substantially, but the exact constant is not determined.

Theorem III (printed p. 31). If (1) has no solution, then

$$\liminf_{x\to\infty}A(x)\,x^{-(\sqrt5-1)/2}<\infty. \tag{13}$$

The theorem shows that Theorem I cannot be improved for every $x$ but that a much sharper inequality holds for infinitely many $x$.

Construction (printed pp. 32–33). There is a sequence with (1) unsolvable and $A(x)>cx^{2/7}$ for every $x$: put $a_1=1$; when $a_1,\ldots,a_{k_i}$ are chosen, put $B_{i+1}=2\sum_{r\le k_i}a_r$ and

$$a_{k_i+l}=1+lB_{i+1},\qquad1\le l\le\Bigl[\frac{B_{i+1}^2}{10}\Bigr], \tag{16}$$

so that $a_{k_{i+1}}\le B_{i+1}^3/10+1$. The paper then asks (printed p. 33) for the supremum $\beta$ of the $\alpha$ for which some sequence with (1) unsolvable has $A(x)>cx^\alpha$ for every $x$, and records $2/7\le\beta\le(\sqrt5-1)/2$.

Remarks printed with the theorems: density $0$ also follows when (1) has only finitely many solutions (p. 29); if (1) is excluded only for $2\le j\le r$ summands, the upper density is at most $1/r$, and $a_k=1+kr$ shows this cannot be sharpened (p. 29); Theorem I is best possible in the sense that for $f(x)\to\infty$ arbitrarily slowly there is a sequence with (1) unsolvable and $A(x)>x/f(x)$ for infinitely many $x$ (pp. 29–30).

Source. P. Erdős, Számelméleti megjegyzések, III. Néhány additív számelméleti problémáról, Mat. Lapok 13 (1962), 28–38 (Hungarian; Russian and English summaries on printed pp. 37–38); printed p. $n$ is PDF p. $n-27$ of the eleven-page scan read for this page. Theorems I–III on printed pp. 28, 30 and 31 (PDF pp. 1, 3 and 4), the construction on pp. 32–33 (PDF pp. 5–6), the English summary on p. 38 (PDF p. 11), all read on the page images; the prose is rendered here in the corpus's words, and the displays keep the paper's numbering in modern notation ((13) is printed with $x=\infty$ under the liminf and as $A(x)/x^{(\sqrt5-1)/2}$).

Read depth. Claims checked: the three theorem statements, the construction (16) and the question on $\beta$ were read clause by clause on the page images, and the English summary (p. 38) was compared with them. The proofs were read for their structure only and are not checked or reconstructed here.

Proof pointer

Theorem I (pp. 28–29): the shifted sequences $A_r=\{a_1+\cdots+a_r+a_k:k>r\}$, $r\ge0$, are pairwise disjoint when (1) is unsolvable, which gives (2) $x\ge\sum_{i\le k}A_i(x)\ge(k+1)A_k(x)$ and (4) $A(x)\le x/(k+1)+\sum_{i\le k}a_i+k$ for every $k$. Theorem II (pp. 30–31): the indices $j$ are split by whether $A(2^{j+1})-A(2^j)\le2^j/j^2$; the first class contributes less than $\sum1/j^2<2$ (display (5)), and for the second class the distinct sums $a_1+\cdots+a_r+a_k$ bound $A(2^{j_r+1})$ from above (displays (6)–(9)), giving (10)–(12) and the total $103$. Theorem III (pp. 31–32): if (13) failed then $a_k=o(k^{(1+\sqrt5)/2})$, and displays (14)–(15) with (2) give a contradiction. The construction's sum-freeness (pp. 32–33) is checked by comparing residues modulo $B_{i+1}$, and the lower bound $A(x)>cx^{2/7}$ follows from (19). An English outline of the Theorem II argument, with an unspecified absolute constant in place of $103$, is Theorem 2 of Benkoski and Erdős.

Dependencies

None; the arguments are elementary. Footnote 1 (p. 29) refers to Problem 4268 of the American Mathematical Monthly for the sharpness example.

Bears on

• Problem 876: the density-zero theorem the site attributes to this paper; the reciprocal-sum bound $103$ that opens the reciprocal-sum question of the site's commentary (Erdős's later restatements print $103$ in 1975 and $100$ in 1977); Theorem III and the $x^{2/7}$ construction as the 1962 bounds on how dense such a sequence can be, both superseded on the density side by Łuczak and Schoen's Theorem 3 and construction. The paper says nothing about the gaps $a_{n+1}-a_n$ the problem asks about beyond what its density statements imply.