---

Statement

Notation (p. 101): $A(x)$ and $A_2(x)$ count the elements of $A$ and of $A+A$ below $x$; a basis is a basis of some order $h$, a set of natural numbers such that every sufficiently large integer is a sum of at most $h$ of its elements.

Conjecture 1 (p. 102, quoted as posed). "If $A$ is a basis and $A(x)=o(x)$, then $A_2(2x)/A(x)\to\infty$."

The introduction (p. 101) announces it as "a modified form of the Erdős—Graham conjecture that has more chance to be true". The paper motivates it by its own example of Theorem 1: there $A(x)$ grows suddenly on a short interval, while sums of two numbers near $x$ lie near $2x$, so $A_2(x)$ grows only later. It proves the threefold analogue, Theorem 2.

Conjecture 2 (p. 102), recorded here because the paper ties it to Conjecture 1: for a finite set $X$ of integers with $\lvert X\rvert=n$ and $\lvert2X\rvert=sn$, every $k$ should satisfy $\lvert kX\rvert\le f(s,k)n$ with $f(s,k)$ depending only on $s$ and $k$ (the print says "depending only on $c$ and $k$" [sic]). The paper says Conjecture 1 could be deduced from it in the same way as Theorem 2 from Theorem 3, that both can probably be deduced from Freiman's main theorem (1966) with $f(s,k)=\exp cks$, and that the authors think the true order of $f(s,k)$ is something like $s^{ck}$.

Source. I. Z. Ruzsa and S. Turjányi, A note on additive bases of integers, Publ. Math. Debrecen 32 (1985), 101--104; Section 3, p. 102, read on the page image of the copy identified on the source card.

Read depth. Claims checked: the conjecture and Conjecture 2 were read clause by clause on the page image. The paper proves neither.

Bears on

• Problem 337: the problem asks for $\lvert(A+A)\cap\{1,\ldots,N\}\rvert/\lvert A\cap\{1,\ldots,N\}\rvert\to\infty$, the Erdős--Graham form that Theorem 1 disproves; the conjecture is the paper's modification, counting $A+A$ up to $2x$. The problem page records, from the formal-conjectures statement file, that this form follows from the Plünnecke--Ruzsa inequality; the 1985 paper itself leaves it open.