Larsen 2026 three questions erdos nathanson asymptotic bases

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theorem_1: Larsen's theorem that for asymptotic bases of order two each of the eight combinations of a divergent representation function, a splitting into two disjoint bases and a minimal subbasis occurs.

theorem_2: Larsen's construction theorem that for any admissible selection mechanism there are disjoint sets B and C, built on the intervals between the powers N_k of four, each with at least floor(n/10^8) balanced representations of every large n other than the N_i, whose sums equal to N_{k+1} all meet F_k for large k.

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Daniel Larsen, Three Questions of Erdős-Nathanson on Asymptotic Bases of Order 2. arXiv preprint (2026). arXiv:2603.03472. The copy read for this card is arXiv version v1 (3 March 2026). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2603.03472), every other right reserved.

Theorems 1 and 2 below were checked against the full text of that copy. Larsen studies three robustness properties of an asymptotic basis A of order 2, with r_A(n) counting the representations n = a + a' with a <= a' in A: (P1) r_A(n) tends to infinity; (P2) A is the union of two disjoint asymptotic bases; (P3) A contains a minimal asymptotic basis. Erdős and Nathanson had shown that r_A(n) > C log n for all large n, with a constant C > 1/log(4/3), gives both (P2) and (P3). Theorem 1 shows that the three properties are independent: each of the eight combinations of truth values occurs for some asymptotic basis. All eight cases come from one construction (Theorem 2) on the intervals between the integers N_i = 4^{i+1}. The case with (P2) true and (P3) false answers no to Question 4 of Erdős and Nathanson's 1988 paper, problem 869. The case with (P1) true and (P2) false gives another negative answer to their Question 2, problem 871, which the paper says the author had answered before. The case with (P1) true and (P3) false answers no to the first question of problem 868; the paper credits that answer, and the stronger growth r_A(n) > epsilon log n of the problem's second question, to the preprint of D. Larsen and M. Larsen, and does not prove that growth bound here. The paper was first read for problem 326, which asks for a minimal basis with a_k/k^2 tending to a non-zero constant; nothing in it concerns that growth condition.

Source: https://arxiv.org/abs/2603.03472.

Bears on. #868 (Theorem 1's cases with (P1) true and (P3) false answer its first question no; the paper credits that answer, with the epsilon log n growth of its second question, to the Larsen and Larsen preprint), #869 (Theorem 1's cases with (P2) true and (P3) false answer it no), #871 (Theorem 1's cases with (P1) true and (P2) false answer it no; the paper says the author had answered it before, with a construction based on Erdős and Nathanson's 1988 paper)

Results.

• Theorem 1 (p. 1): for each of the eight assignments of true or false to (P1), (P2) and (P3) there is an asymptotic basis of order 2 with exactly those properties.
• Theorem 2 (p. 2): let $N_k=4^{k+1}$ and $h(n)=\lfloor10^{-8}n\rfloor$. For any selection mechanism whose outputs $G_k$, $H_k$ are disjoint with no element of $F_k=G_k\cup H_k$ greater than $N_k$, there are sets B and C, built from parts $B_k$, $C_k$ strictly between $N_k$ and $N_{k+1}$ together with the reflected sets $N_{k+1}-G_k$ and $N_{k+1}-H_k$, such that $B\cap C=\emptyset$; each of B and C gives at least $h(n)$ representations of n with summand ratio in [1,100] for all sufficiently large n not among the $N_i$; and for large k every representation of $N_{k+1}$ as a sum of two elements of $B\cup C$ meets $F_k$.

Read status. Claims checked: Theorems 1 and 2 against arXiv v1; the proofs were read for their structure only.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.