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Statement

Question (p. 853, unnumbered, quoted). "Does there exist a sequence $b_1<b_2<\cdots$ satisfying $N(b_j,x)<c'_{10}x/\log x$ so that every sufficiently large integer is of the form $2^l+b_j$?"

Here $N(b_j,x)$ counts the $b_j\le x$. The paper introduces the question as seeming more difficult than its constructions for squares and $k$-th powers on the same page (see the remark on p. 853), and does not answer it.

Source. P. Erdős, Some results on additive number theory, Proc. Amer. Math. Soc. 5 (1954), 847-853, p. 853. The edition read is identified on the source card.

Read depth. Claims checked: the question was read clause by clause on the printed page. Nothing here is independently reviewed.

Proof pointer

None: the paper poses the question without a proof or a sketch.

Dependencies

None.

Bears on

• Problem 221: the problem asks whether some $A\subset\mathbb N$ with $\lvert A\cap\{1,\ldots,N\}\rvert\ll N/\log N$ for all large $N$ has every large integer of the form $2^k+a$. That is the question posed here; this page records the 1954 posing only, and the answers are on the problem page.