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Updated Oct 8

Problem 28

Statement. If $A\subseteq \mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\limsup 1_A\ast 1_A(n)=\infty$ .

Status. Open.

Source. erdosproblems.com/28, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #28, https://www.erdosproblems.com/28.

References.

[Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; section C9 "Packing sums of pairs", printed p. 177: the Erdős--Turán conjecture that $f(n)>0$ for all large $n$ forces $\limsup f(n)=\infty$ , with the prize offer. Library home: guy_2004_unsolved_problems_number_theory.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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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 (33)

Additive Bases and Sidon Sets Additive Bases and Sidon Sets Borwein et al.: An old conjecture of Erdős–Turán on additive bases Conjecture 0.1 (p. 475): an everywhere-positive square of a 0-1 series has unbounded coefficients (Erdős–Turán)Conjecture 2.1 (p. 476): eventual positivity of f(z)^2 forces unbounded coefficients (Erdős–Turán, original version)Theorem 1.1 (pp. 475--476): an everywhere-positive square f(z)^2 with nonnegative integer coefficients has a coefficient at least 8 Theorem 2.7 (p. 479): if the prefix set E(k) is finite, no basis has all representation counts at most k additive_bases/erdos_1941_problem_sidon_additive_number_theory_related Conjecture (2) (p. 215): if every large n is a sum a_i + a_j, the representation counts are unbounded Grekos et al.: On the Erdős–Turán conjecture Corollaries 3.4, 3.10 and 3.17 (pp. 343–345): tau, alpha and sigma give further equivalent forms of the Erdős–Turán conjecture Lemma 2.2 (p. 342): the Diagonal Lemma, with Corollary 2.3 on diagonals of finite bases 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 Nathanson: Generalized additive bases, König's lemma, and the Erdős–Turán conjecture Theorem 1 (p. 3): a finite basis of order H exists exactly when max(H_n)/n has positive liminf Theorem 4 (p. 6): an R-basis of order H exists exactly when arbitrarily large finite R-bases do Theorem 6 (pp. 7–8): a basis of order h with bounded representations exists exactly when arbitrarily large finite ones do additive_bases/nathanson_2014_paul_erdos_additive_bases Conjecture (p. 3, unnumbered): the Erdős-Turán conjecture, recorded as open in 2014 additive_bases/obryant_2004_complete_annotated_bibliography_work_related_sidon Question (p. 17): must a bounded A^* vanish infinitely often additive_bases/pliego_2024_erdos_turan_conjecture_growth_b_2 Conjecture 1.1 (p. 1): no sequence A and no fixed g >= 2 have 1 <= r_A(n) <= g for every large n (Erdős-Turán)additive_bases/ruzsa_1990_just_basis Theorem 1: a set in [0, 3p^2] of size at most 12p whose sumset covers [2p^2, 4p^2] with at most 288 representations Theorem 2: a basis of order 2 whose representation counts are bounded in square mean additive_bases/sarkozy_1997_additive_representation_functions additive_combinatorics/erdos_1956_problems_results_additive_number_theory Conjecture (p. 128): the Erdős–Turán conjecture and the stronger conjecture under a_k < ck^2 discrete_geometry/erdos_1981_applications_graph_theory_combinatorial_methods_number The Erdős–Turán conjecture and its multiplicative analogue, pp. 144-145, with the Nešetřil–Rödl proof number_theory/guy_2004_unsolved_problems_number_theory

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