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Statement

Take $A$, $B$, $A^+$ and $B^+$ as in Theorem 3: a partition of the positive integers into two classes and the sets of sums of distinct elements of each class. For an infinite sequence $a_1<a_2<\cdots$ the paper (p. 38) defines the upper logarithmic density

$$\bar d_\ell(A)=\limsup_{x\to\infty}\frac{1}{\log x}\sum_{a_i\le x}\frac{1}{a_i}.$$

Display (23) (p. 38). There are partitions for which

$$\max\bigl(\bar d_\ell(A^+),\bar d_\ell(B^+)\bigr)<1. \tag{23}$$

Display (24) (p. 39). The example takes for $A$ the integers $m$ with

$$2^{4^{2k}}<m\le2^{4^{2k+1}},\qquad k=0,1,\ldots, \tag{24}$$

and for $B$ the complementary set; the print writes the index range as "$m=0,1,\ldots$" [sic]. Erdős states that a simple computation, which he does not give, shows that this partition satisfies (23).

Display (25) (p. 39). Erdős is sure that every partition satisfies

$$\max\bigl(\bar d_\ell(A^+),\bar d_\ell(B^+)\bigr)>\tfrac12, \tag{25}$$

and expects this to be not difficult to prove; the paper gives no proof.

Display (26) (p. 39). He writes that he does not see how to determine

$$\min_{A,B}\max\bigl(\bar d_\ell(A^+),\bar d_\ell(B^+)\bigr)=c, \tag{26}$$

and that the proof of (26) is perhaps not so trivial, while the proofs of Theorem 3 and probably of (25) are routine.

Source. P. Erdős, Miscellaneous problems in number theory, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congr. Numer. 34 (1982), 25--45; Part II, the definition and display (23) on p. 38, displays (24)--(26) on p. 39. The edition read is identified on the source card.

Read depth. Claims checked: the passage was read clause by clause on the page images. The computation behind (23) is not given in the paper and was not re-derived here; (25) and (26) are an expectation and a question.

Dependencies

None.

Bears on

• Problem 1211: display (26) is the problem's question, the least value over two-class partitions of the larger upper logarithmic density of the two subset-sum sets. The example (24), for which Erdős asserts (23) without giving the computation, would put the value below $1$, and (25) is his expectation that it exceeds $1/2$. The paper records no determination of the value. The problem page records the value $(2+\sqrt3)/4$ found by Conlon, Fox and Pham, through their Theorem 1.