## $$B_2[2]$$: The vise tightens. (Ensembles $$B_2[2]$$: L’étau 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

### Keywords:

$$B_2[2]$$-sets; maximal cardinality; asymptotic bounds
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