The postage stamp problem and arithmetic in base $$r$$.(English)Zbl 1174.11013

Summary: Let $$h,k$$ be fixed positive integers, and let $$A$$ be any set of positive integers. Let $$hA:=\{a_1+a_2+\cdots +a_r\: a_i \in A, r \leq h\}$$ denote the set of all integers representable as a sum of no more than $$h$$ elements of $$A$$, and let $$n(h,A)$$ denote the largest integer $$n$$ such that $$\{1,2,\ldots ,n\} \subseteq hA$$. Let $$n(h,k):=\max _A\:n(h,A)$$, where the maximum is taken over all sets $$A$$ with $$k$$ elements. We determine $$n(h,A)$$ when the elements of $$A$$ are in geometric progression. In particular, this results in the evaluation of $$n(h,2)$$ and yields surprisingly sharp lower bounds for $$n(h,k)$$, particularly for $$k=3$$.

MSC:

 11B13 Additive bases, including sumsets 11D04 Linear Diophantine equations
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References:

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