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Keywords:
$h$-basis; extremal $h$-basis; geometric progression
$h$-basis; extremal $h$-basis; geometric progression
Summary:
Let $h,k$ be fixed positive integers, and let $A$ be any set of positive integers. Let $hA:=\{a_1+a_2+\cdots +a_r\colon a_i \in A, r \le h\}$ 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 $n(h,k):=\max _A\colon n(h,A)$, 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 $k=3$.
References:
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