Larsen 2026 robust additive bases without minimal subbases

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theorem_1: Daniel and Michael Larsen's theorem that some set of positive integers has more than epsilon log m representations of every large m as a sum a + b with a <= b, for a fixed epsilon > 0, yet contains no minimal additive subbasis of order 2; the construction gives epsilon = 15/(512 log 2).

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Daniel Larsen, Michael Larsen, Robust additive bases without minimal subbases. arXiv preprint (2026). arXiv:2601.18507. The copy read for this card is arXiv version v1 (26 January 2026). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2601.18507), every other right reserved.

Here $r_A(m)$ counts the pairs $(a,b)\in A^2$ with $a+b=m$ and $a\le b$, and an additive basis has order $2$ throughout (p. 1). The paper recalls that Erdős and Nathanson proved that $A$ must contain a minimal subbasis when $r_A(m)>c\log m$ for all sufficiently large $m$ with $c>\log(4/3)^{-1}$, and says they conjectured that this is not true for all positive $c$ (p. 1). Theorem 1 (p. 1) proves that conjecture: there are $\varepsilon>0$ and $A\subset\mathbb N$ with $r_A(m)>\varepsilon\log m$ for all sufficiently large $m$ such that $A$ contains no minimal additive subbasis of order $2$. The remark after Lemma 10 gives $\varepsilon=\frac{15}{512\log2}\approx0.042$, not optimized (p. 8).

The construction is random and runs by generations on the intervals $I_n=[X_n,X_{n+1})$, $X_n=2^{2^n}$. Each $m$ is put in a random set $A_n$ independently with probability $\min\bigl(1,40\sqrt{\log m/m}\bigr)$. Lemma 2 (p. 2) gives, with probability $1$ for all but finitely many $n$ and for every $m\in I_n$, more than $160\log m$ representations of $m$ from $A(n)\cap[X_n/4,X_{n+1})$, where $A(n)=A_1\cup\cdots\cup A_n$, and fewer than $c\log X_n$ from $A(n)$, for an absolute constant $c$; the proof uses Chernoff bounds and the Borel--Cantelli lemma. Proposition 5 (p. 5) rules out, almost surely for large $n$, $k\ge17$ pairwise distinct triples in $A(n)$ with one common value of $x_i+y_i$ and another of $y_i+z_i$, both in $I_n$. Section 3 (pp. 6--9) then chooses a random set $B_n$ in $[X_{n+1}/6,X_{n+1}/4)$, deletes every summand of its elements, and adds new elements so that each $b\in B_n$ is represented only in a prescribed way (Lemma 10, p. 8). The paper describes the sets $B_n$ as generalizing the single integers $N_n$ of Erdős and Nathanson's 1989 construction, whose representation counts are bounded; having many fragile elements per generation is what lets the counts of the elements of $B_n$ grow logarithmically (p. 2). The paper works in order $2$ only.

Source: https://arxiv.org/abs/2601.18507.

Read status: claims checked for Theorem 1, the constant on p. 8 and the statements of Lemma 2 and Proposition 5, read clause by clause on the print; the proof was followed for structure only. Nothing here is independently reviewed. Result page: theorem_1.

Bears on. #868: Theorem 1 (p. 1) gives an additive basis of order $2$ with $r_A(m)>\varepsilon\log m$ for all large $m$, $\varepsilon=\frac{15}{512\log2}$, that contains no minimal additive basis of order $2$; since $1_A\ast1_A(m)\ge r_A(m)$, this answers both of the problem's questions no. #870: the paper treats order $2$ only and states nothing about bases of order $k\ge3$.

Results.

• Theorem 1 (p. 1): there are $\varepsilon>0$ and $A\subset\mathbb N$ with $r_A(m)>\varepsilon\log m$ for all sufficiently large $m$ such that $A$ contains no minimal additive subbasis of order $2$.

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