---

Statement

An additive system is a family $(B_j)_{j\in J}$ of sets of integers with $0\in B_j$ and $|B_j|\ge2$ for all $j$, such that the sums $\sum_{j\in J}b_j$ with $b_j\in B_j$ for all $j$ and $b_j\ne0$ for only finitely many $j$ are exactly the nonnegative integers, and every nonnegative integer has exactly one such representation (p. 1).

Lemma 2 (p. 3). Let $\mathcal B=(B_j)_{j\in J}$ be an additive system, and let $\{J_i\}_{i\in I}$ be a partition of $J$ into pairwise disjoint nonempty sets. If

$$A_i=\sum_{j\in J_i}B_j$$

for each $i\in I$, then $\mathcal A=(A_i)_{i\in I}$ is an additive system.

Definitions that follow it (pp. 3--4). A system $\mathcal A$ obtained from $\mathcal B$ in this way is a contraction of $\mathcal B$; the paper notes that de Bruijn called it a degeneration. The index set $I$ may be finite or infinite. Every additive system is a contraction of itself, and taking $I=\{1\}$ and $J_1=J$ shows that the one-set system $(\mathbf N_0)$ is a contraction of every additive system. A contraction is proper if at least one $A_i$ is the sum of at least two sets of $\mathcal B$.

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 on p. 3, the definitions after it on pp. 3--4.

Read depth. Claims checked: the statement and the definitions were read clause by clause on the page images of pp. 1, 3 and 4. Nothing here is independently reviewed.

Proof pointer

The paper gives no written proof; it says (p. 3) that the lemma follows immediately from the definition of an additive system. A representation of $n$ in the system $\mathcal A$ expands, set by set, into a representation in $\mathcal B$, and conversely a representation in $\mathcal B$ regroups along the blocks $J_i$; uniqueness in $\mathcal B$ therefore gives uniqueness in $\mathcal A$.

Dependencies

None.

Bears on

• Problem 1145, as the source of a model case only. Applied to the binary system $(\{0,2^{i-1}\})_{i\in\mathbf N}$ (the paper's Example 2, p. 2) with the positions split by parity, the lemma gives $\mathbf N_0=U\oplus V$ with $U$ the sums of distinct powers $4^j$ and $V=2U$; the source card works out that the translates $U+1$ and $V+1$ give every $n\ge2$ exactly one representation while their $n$th elements have ratio tending to $1/2$, not $1$. This derivation is the card's, not the paper's; the paper does not mention the problem, and the pair does not meet the problem's hypothesis $a_n/b_n\to1$.