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Statement

Setting (p. 2). For $A\subseteq\mathbb Z$, $r_A(n)$ is the number of pairs $(a',a'')\in A\times A$ with $a''-a'=n$ (the paper prints the set of pairs). A set $A\subseteq\mathbb Z$ is a perfect difference set if every non-zero integer has a unique representation as a difference of two elements of $A$; in the paper's terms, $r_A(n)=1$ for every $n\in\mathbb N$. $\mathbb N$ is the set of positive integers.

Theorem 3 (p. 2, quoted). "There is a partition $\mathbb N=\cup_{k=1}^{\infty}A_k$ of the set of all positive integers such that each $A_k$ is a perfect difference set and $|A_i\cap(A_j+z)|\le2$ for any $i,j,z\in\mathbb N$."

The intersection condition is the paper's way of making the parts have "completely different structure" (p. 2): no three elements of any part reappear, shifted by a positive integer, in the same or another part. By contrast, the paper remarks (p. 2) that for any finite partition of $\mathbb N$ some part $A$ has $r_A(n)=\infty$ for arbitrarily large $n$.

Source. Vsevolod F. Lev, Reconstructing integer sets from their representation functions, Electron. J. Combin. 11 (2004), no. 1, Research Paper 78, 6 pp., doi:10.37236/1831: the statement on p. 2, the proof on pp. 3--4 (Section 2, pp. 3--4). The edition read is identified on the source card.

Read depth. Claims checked: the definitions and the statement were read clause by clause on the printed pages. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 3--4. A greedy construction in steps. A function $f:\mathbb N\to\mathbb N$ with $f(2m-1)\le m$ and $f(2m)=m+1$ (the paper's (3)), and with every fibre $f^{-1}(k)$ infinite, schedules which part is extended at each step. At an even step $2m$ the still empty part $A_{m+1}$ receives the least positive integer not yet used. At an odd step $n$ the part $A_k$, $k=f(n)$, receives $z$ and $z+d$, where $d$ is the least positive integer not yet a difference in $A_k$ and $z$ is chosen so that both numbers are new, no non-trivial equation $a_1-a_2=a_3-a_4$ arises in $A_k$, and no new triple in $A_k$ is a translate of a triple in another part. Each condition excludes only finitely many $z$; for the last one the paper uses that at step $n$ all but $(n+1)/2$ parts are still empty.

Dependencies

No other result of the paper.

Bears on

No Erdős problem in this corpus directly. The single-set simplification of this construction, the greedy perfect difference set, bears on Problem 1194.