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Let $h, k$ be fixed positive integers, and let $A$ be any set of positive integers. Let $h A : = {a_{1} + a_{2} + \dots + a_{r} : a_{i} \in A, r \leq h}$ 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 ${1, 2, ..., n} \subseteq h A$ . 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 $k = 3$ .

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The additive complements of primes and Goldbach's problem

The class of Erdős-Turán sets

The Erdős-Heilbronn problem for finite groups

The Erdős-Turán property for a class of bases

The inverse problem for representation functions for general linear forms.

The minimal range of additive h-bases.

The postage stamp problem and arithmetic in base $r$

The prime power conjecture is true for $n < 2, 000, 000$ .

The structure of maximal zero-sum free sequences

Théorie additive des nombres. Densité de Schnirelman

Two remarks on linear forms in non-negative integers.

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