Erdős Problem #339

Let $A\subseteq \mathbb{N}$ be a basis of order $r$. Must the set of integers representable as the sum of exactly $r$ distinct elements from $A$ have positive lower density?

#339: [ErGr80,p.52]

number theory | additive basis

Erdős and Graham also ask whether if the set of integers which are the sum of $r$ elements from $A$ has positive upper density then must the set of integers representable as the sum of exactly $r$ distinct elements have positive upper density?

The answer to both questions is yes, as proved by Hegyvári, Hennecart, and Plagne [HHP03].

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This page was last edited 14 October 2025.

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