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Statement

Setting (Section 2, p. 47). Let $a_1,\ldots,a_{2n}$ be distinct integers with $1\le a_i\le4n$, and let $b_1,\ldots,b_{2n}$ be the remaining integers of the interval $(1,4n)$ (the complement of the $a_i$ among the integers up to $4n$).

Question (p. 47). Is there always an integer $t$ for which the number of solutions of $a_i+t=b_j$ is at least $n$? The paper says it could not decide this, and notes that the choice $a_i=n+i$, $1\le i\le2n$, shows that $n$ would be best possible.

Theorem (p. 47). There is always an integer $t$ for which the number of solutions of $a_i+t=b_j$ is at least $n/2$.

Reported result (p. 47). Scherk (the paper's reference [3]) proved that for a suitable $t$ the number of solutions is at least $(2-\sqrt2)n$.

The English summary (p. 48) states the theorem as: "I show that there exists an integer $x$ so that there are at least $n/2$ $b$'s among the integers $a_i+x$. Scherk improved this to $(2-\sqrt{2})n$. It is not known whether this can further be improved to $n$."

Proof pointer

Averaging (p. 47): over the shifts $-4n<x<4n$ the equation $a_i+x=b_j$ has $4n^2$ solutions in all, one for each pair $(a_i,b_j)$, and there are fewer than $8n$ shifts, so some shift carries at least $n/2$ of them.

Read depth. Claims checked: the setting, the question, the theorem, the example $a_i=n+i$ and Scherk's bound were read clause by clause on the page images of pp. 47 and 48; the averaging was re-derived here.

Source. P. Erdős, Some remarks on number theory (in Hebrew), Riveon Lematematika 9 (1955), 45--48; the edition read is named on the source card.

Dependencies

None.

Bears on

• Problem 36: with $N=2n$, the $a_i$ and $b_j$ form a partition of $\{1,\ldots,2N\}$ into two halves of size $N$, and the count of solutions of $a_i+t=b_j$ is the problem's count of differences $b-a=t$ (the problem counts $a-b=x$, which is the same quantity with the halves' names exchanged). In the problem's normalization the theorem is $c\ge1/4$, Scherk's bound is $c\ge1-1/\sqrt2$, and the paper's question asks whether $c=1/2$ is admissible, which the example $a_i=n+i$ shows would be optimal; the paper poses this as a question it could not decide, not as a conjecture. The paper treats only totals $4n$, that is even $N$, and proves nothing beyond the bound $1/4$.