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\(B_2[2]\): The vise tightens. (Ensembles \(B_2[2]\): L’√©tau se resserre.) (French) Zbl 1009.11007

A \(B_h[g]\) set is a set of integers with the property that any number has at most \(g\) representations as a sum of \(h\) elements of this set. \(F_h(N,g)\) denotes the size of a maximal \(B_h[g]\) subset of \(\{1, 2, \dots , N\}\). It is a classical result that \(F_2(N,1)\sim \sqrt N\) (Sidon sets), and no asymptotic result is known for any other \(h,g\). This paper improves the best known numerical estimates for \(F_2(N,2)/\sqrt N\). The new lower bound is \(4/\sqrt 7+o(1)\) and the upper bound is 2.3218.

MSC:

11B34 Representation functions
11B75 Other combinatorial number theory
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