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Statement

Setting (p. 1). For $A\subseteq\mathbb Z$ and $n\in\mathbb Z$, $R_A^{(3)}(n)$ counts the pairs $(a_1,a_2)\in A\times A$ with $a_1+a_2=n$ and $a_1\le a_2$. The paper prints the defining set of pairs and compares these functions as counts. $\mathbb N$ is the set of positive integers.

Theorem 2 (Chen and Wang [CW03], p. 2). Define $T:\mathbb N\to\{1,-1\}$ by

$$T(1)=1,\qquad T(2n)=-T(2n-1),\qquad T(2n+1)=-T(n+1)\qquad(n\in\mathbb N),$$

and put $A=\{n\in\mathbb N:T(n)=1\}$, $B=\{n\in\mathbb N:T(n)=-1\}$. Then $R_A^{(3)}(n)=R_B^{(3)}(n)$ for all integer $n\ge3$.

The only change from Dombi's Theorem 1 is the sign in the recursion at odd arguments, and the range is $n\ge3$ rather than all $n\in\mathbb N$. The theorem is Chen and Wang's (Y.-G. Chen and B. Wang, On the additive properties of two special sequences, Acta Arith. 110 (2003), no. 3, 299--303); the paper gives it a new proof common with Theorem 1.

Footnote 1 (p. 2). Dombi had conjectured that no $A,B\subseteq\mathbb N$ with infinite symmetric difference satisfy $R_A^{(3)}(n)=R_B^{(3)}(n)$ for $n$ large enough; Theorem 2 shows that such sets exist.

Essential uniqueness (p. 3, unnumbered). For a partition $\mathbb N=A\cup B$ with $T(n)=1$ on $A$ and $T(n)=-1$ on $B$, the paper says its proof shows that $R_A^{(3)}(n)=R_B^{(3)}(n)$ for all sufficiently large $n$ if and only if $T(1)+\cdots+T(2n)=0$ and $T(2n)=T(n)$ for all but finitely many $n\in\mathbb N$. It states, leaving the check to the reader, that this is equivalent to the existence of $n_0\in\mathbb N$ with $T(2n)=-T(2n-1)$ and $T(2n-1)=-T(n)$ for $n\ge n_0$, and $T(1)+\cdots+T(2n_0)=0$.

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 and footnote 1 on p. 2, the common proof of Theorems 1 and 2 and the uniqueness remark on p. 3 (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

P. 3. The argument is that of Theorem 1 with $j=3$: twice the generating series of $R_A^{(3)}$ is $\alpha(x)^2+\alpha(x^2)$, so the difference of the two series becomes a series in $x^{2n}$ with coefficients $-T(2n)+T(n)$. The recursion gives $T(2n)=T(n)$ for every $n\ge2$; it fails only at $n=1$, where $T(2)=-1$ and $T(1)=1$, so the coefficient of $x^2$ is non-zero and $n=2$ is excluded.

Dependencies

No other result of the paper.

Bears on

No Erdős problem in this corpus. The theorem answers, for $R^{(3)}$, a question the paper attributes to Sárközy.