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Statement

Notation: $[0,g)=\{0,1,\ldots,g-1\}$, $g*B=\{gb:b\in B\}$. An additive system is a family of sets of integers, each containing $0$ and at least two elements, whose finite-support sums are exactly the nonnegative integers, each with exactly one representation (p. 1). The dilation of an additive system $\mathcal B=(B_i)_{i\in I}$ by an integer $g\ge2$ adjoins a new index $i_1\notin I$ with set $[0,g)$ and replaces each $B_i$ by $g*B_i$ (p. 2); a contraction groups the sets of a system along a partition of its index set into nonempty blocks and sums each block (Lemma 2, p. 3). "A contraction of $\mathcal B$ dilated by $g$" means the system obtained by first dilating $\mathcal B$ by $g$ and then contracting (p. 4).

Lemma 7 (p. 5). Let $\mathcal A=(A_i)_{i\in I}$ be an additive system with $|I|\ge2$. Then there exist $i_1\in I$, an integer $g\ge2$, and a family of sets $\mathcal B=(B_i)_{i\in I}$ such that

$$A_{i_1}=[0,g)\oplus g*B_{i_1}$$

and $A_i=g*B_i$ for all $i\in I\setminus\{i_1\}$. If $B_{i_1}=\{0\}$, then $\mathcal B=(B_i)_{i\in I\setminus\{i_1\}}$ is an additive system and $\mathcal A$ is the dilation of $\mathcal B$ by $g$. If $B_{i_1}\ne\{0\}$, then $\mathcal B=(B_i)_{i\in I}$ is an additive system and $\mathcal A$ is a contraction of $\mathcal B$ dilated by $g$.

In the proof (pp. 6--7), $i_1$ is the index of the set containing $1$, $g$ is the least positive integer not in $A_{i_1}$, and $B_i=\{k\in\mathbf N_0:kg\in A_i\}$ for every $i\in I$; the decomposition of $A_{i_1}$ is the paper's equation (3) (p. 7).

Source. Melvyn B. Nathanson, Additive systems and a theorem of de Bruijn, Amer. Math. Monthly 121 (2014), no. 1, 5--17, doi:10.4169/amer.math.monthly.121.01.005, read in the arXiv version 1301.6208v2 (12 April 2013) identified on the source card, whose pages are numbered 1 to 12; labels and pages here are that version's. The lemma is stated on p. 5 and proved on pp. 6--7.

Read depth. Claims checked: the statement and the proof were read clause by clause on the page images of pp. 5--7, without a line-by-line check of the induction. In the case $B_{i_1}=\{0\}$ the proof prints $A_{i_1}=[0,g-1)$ (p. 7), where the statement gives $[0,g)$; the statement is the one recorded above. Nothing here is independently reviewed.

Proof pointer

Pp. 6--7. Because $|I|\ge2$ no set is all of $\mathbf N_0$, so the set containing $1$ has a least missing positive integer $g\ge2$, and $[0,g)$ lies in it. Uniqueness forces $g$ itself to lie in another set $A_{i_2}$, and an induction on $k\ge0$ over the blocks $[kg,(k+1)g)$ shows that no other set meets $[kg+1,(k+1)g)$, and that $A_{i_1}$ either contains a whole block $[kg,(k+1)g)$ or misses it. Hence every set other than $A_{i_1}$ consists of multiples of $g$ and $A_{i_1}$ is a union of whole blocks, which gives the displayed decomposition; dividing the representation of $1+gn$ by $g$ shows that $\mathcal B$ is again an additive system.

Dependencies

The definitions of dilation (p. 2) and of contraction (Lemma 2, p. 3).

Bears on

No Erdős problem directly. Repeated application of the lemma supplies the radices $g_1,g_2,\ldots$ in the proof of Theorem 3.