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Some elementary results are given on restricted sums 2 ∧ A and 3 ∧ A of a subset A ⊂ Z/nZ, in the case when the cardinality of A is close to n/2. In the last section, some of these results are used to derive some new values of a function related to the Erdős-Ginzburg-Ziv problem.

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I give an account of results and open problems on asymptotic bases in general additive number theory, inspired and arisen from the paper "On bases with an exact order" by Paul Erdős and Ronald L. Graham, published in Acta Arith. 37 (1980), 201-207. I survey papers by Melvyn B. Nathanson, Xing-De Jia, John C. M. Nash, Alain Plagne, Julien Cassaigne, Bruno Deschamps and myself.-To Ron Graham, with my warmest wishes for his 70th birthday. 1. The Original Erdős-Graham Question I present here the original Erdős-Graham question and I discuss two observations that allow us to extend their setting.