Abstract
Let Fh(N) be the maximum number of elements that can be selected from the set {1, …, N} such that all the sums a1+…+ah, a1⩽…⩽ah are different. We introduce new combinatorial and analytic ideas to prove new upper bounds for Fh(N). In particular we proveF3(N)⩽41+16/(π+2)4N1/3+o(N1/3),F4(N)⩽81+16/(π+2)4N1/4+o(N1/4).Besides, our techniques have an independent interest for further research in additive number theory.
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