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Statement

Setting (p. 295). For a set $A=\{a_i\}$ of nonnegative integers, $k(x)$ is the number of $a_i$ less than $x$.

Problem 12 (p. 295). Erdős asks three questions.

1. If every natural number $n$ has a representation $n=a_i+2^j$, it is known (Lorentz, [36, Theorem 2] of the paper) that such a set $A$ exists with $k(x)<cx(\log\log x)/\log x$. Can the factor $\log\log x$ be dropped?
2. There is a set $\{a_i\}$ such that every natural number has a representation $n=a_i+p_j$ with $p_j$ prime and $k(x)<c(\log x)^2$ (Erdős, [18, p. 847] of the paper). Can this be improved?
3. Hanani's question (oral communication). Let $\{a_i\}$ and $\{b_j\}$ be two increasing sequences of natural numbers such that every $n$ has a representation $n=a_i+b_j$. Is it true that (quoted) "$\limsup \frac{k_a(x)\,k_b(x)}{x} > 1$?" The paper does not define $k_a$ and $k_b$; by the setting above they count the terms of each sequence below $x$.

The paper poses the three questions and resolves none of them.

Source. P. Erdős, Some unsolved problems, Michigan Math. J. 4 (1957), 291--300; §A, Problem 12, p. 295. The edition read is identified on the source card.

Read depth. Claims checked: the item was read clause by clause on the page images of the journal print. The two known constructions are cited, not proved here.

Dependencies

None.

Bears on

• Problem 785: the problem takes Hanani's setting, with $A+B$ required to contain all large integers rather than every $n$, and asks about the case $A(x)B(x)\sim x$, where Hanani's lim sup equals $1$: must $A(x)B(x)-x$ then tend to infinity? The paper records only Hanani's question and does not resolve it.