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Updated Oct 8

Problem 1145

Statement. Let $A=\{1\leq a_1<a_2<\cdots\}$ and $B=\{1\leq b_1<b_2<\cdots\}$ be sets of integers with $a_n/b_n\to 1$ .

If $A+B$ contains all sufficiently large positive integers then is it true that $\limsup 1_A\ast 1_B(n)=\infty$ ?

Formulation. The site's wording as accessed. On this page $1_A\ast 1_B(n)$ is read as the number of ordered pairs $(a,b)\in A\times B$ with $a+b=n$ , so the question asks whether, once every sufficiently large positive integer has such a representation, the representation count is unbounded; this reading is made here, not on the site.

Status. Open.

Source. erdosproblems.com/1145, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1145, https://www.erdosproblems.com/1145.

Formalization. Statement in formal-conjectures.

Current assessment

No current assessment is recorded. The status above is imported from the dated site record. This page records no current literature search or independent assessment of proof coverage.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (20)

Additive Bases and Sidon Sets Bárány et al.: Lagrange-like spectrum of perfect additive complements Borwein et al.: An old conjecture of Erdős–Turán on additive bases Theorem 1.1 (pp. 475--476): an everywhere-positive square f(z)^2 with nonnegative integer coefficients has a coefficient at least 8 Fang–Sándor: On function $SX$ of additive complements Fang–Sándor: On sets with sum and difference structure Grekos et al.: On the Erdős–Turán conjecture Lemma 2.2 (p. 342): the Diagonal Lemma, with Corollary 2.3 on diagonals of finite bases Lemma 5.4 (p. 347): the representation function of A + d*B with d > max A Theorem 2.1 (p. 341): the Erdős–Turán conjecture is equivalent to rho(x) tending to infinity Theorem 6.9 (p. 350): every basis of order two of N has some ordered representation count at least 6 Konstantoulas: Lower bounds for a conjecture of Erdős and Turán Ma: A Note on Additive Complements Narkiewicz: Remarks on a conjecture of Hanani in additive number theory Nathanson: Generalized additive bases, König's lemma, and the Erdős–Turán conjecture Theorem 6 (pp. 7–8): a basis of order h with bounded representations exists exactly when arbitrarily large finite ones do Nathanson: Additive systems and a theorem of de Bruijn Lemma 2 (p. 3): grouping the components of an additive system gives an additive system (contraction)Theorem 2 (p. 5): an infinite radix sequence generates an additive system (British number system)Theorem 3 (p. 7): de Bruijn's theorem, every additive system is a British number system or a proper contraction of one

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