For every integer h≥ 3 and every real number alpha with 0<alpha<1/h, does there exist a set A⊆N_0 such that: (i) hA=a_1+…+a_h:a_1,…,a_h∈ A contains all sufficiently large integers; (ii) for every a∈ A, h(A setminusa) does not contain all sufficiently large integers; and (iii) A has asymptotic density alpha, namely lim_x→∞frac|A cap[0,x]|x=alpha?

Erdős–Nathanson problem on minimal asymptotic bases

For every integer $h\ge 3$ and every real number $\alpha$ with $0<\alpha<1/h$ , does there exist a set $A\subseteq\mathbb{N}_0$ such that: (i) $hA=\{a_1+\cdots+a_h:a_1,\ldots,a_h\in A\}$ contains all sufficiently large integers; (ii) for every $a\in A$ , $h(A\setminus\{a\})$ does not contain all sufficiently large integers; and (iii) $A$ has asymptotic density $\alpha$ , namely $\lim_{x\to\infty}\frac{|A\cap[0,x]|}{x}=\alpha$ ?

References

Primary source

arXiv

Additional references

On a problem of Erdős and Nathanson related to minimal asymptotic bases of order h — arXiv

additive-number-theory

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

An unrefereed preprint claims to settle the existence question for every allowed density, but the result has not been independently verified.

Erdős and Nathanson posed the existence question in 1988 for minimal asymptotic bases of order $h \ge 3$ . The claimed result would complete the density classification for these bases.

Known results

Erdős and Nathanson proved existence at density $1/h$ for every $h \ge 2$ .
They also obtained existence for every $\alpha \in (0,1/(2h-2))$ .
For order $2$ , every density $\alpha \in (0,1/2]$ occurs.

September 8, 2026 claimed solution

A report dated September 8, 2026, points to an unrefereed preprint claiming an affirmative answer for every admissible density when $h \ge 3$ .

Current status (as of September 2026): The classical partial density results are established, while the claimed complete answer for $h \ge 3$ remains unverified.

Sources

No solutions have been posted yet.