Problem 338

Erdős Problem #338: declared status 'open'. Formalized: no. The restricted order of a basis is the least integer $t$ (if it exists) such that every large integer is the sum of at most $t$ distinct summands from $A$. What are necessary and sufficient conditions that this exists? Can it be bounded (when it exists) in terms of the order of the basis? What are necessary and sufficient conditions that this is equal to the order of the basis? Current best: Bateman has observed that for $h\geq 3$ there is a basis of order $h$ with no restricted order, taking$$A=\{1\}\cup \{x>0 : h\mid x\}.$$Kelly [Ke57] has shown that any basis of order $2$ has restricted order at most $4$ and conjectured it always has restricted order at most $3$ (which he proved under the additional assumption that the basis has positive lower density). [He05] Hennecart, Fran\c cois, On the restricted order of asymptotic bases of order two. Prize: no. Tags: additive basis, number theory.