Some remarks on Bh[g] sequences

Abstract An increasing sequence of natural numbers A = ( a i ) (finite or infinite) is called B h [ g ] if every n ϵ N can be written in at most g ways as a sum of h ( h ≥ 2) elements of A . F h ( n , g ) is the cardinality of the largest B h [ g ] sequence in {1,…, n }. A well-known question asks for bounds on F h ( n , g ) (see H. Halberstam and K. Roth, Sequences , Oxford Univ. Press, London/New York, 1966 ). We obtain bounds on F h ( n , g ) in three different ways (using Banach space theory, using estimates on trigonometrical sums, and by using a gap theorem for primes). In turn these bounds on F h ( n , g ) yield some very partial answers to a generalization of a question of Bose and Chowla and of a generalization of a question of P. Erdos and P. Turan [ J. London Math. Soc. 16 (1941) , 212–215]. Bose and Chowla's question is: given natural numbers a 1 a 2 … a n δn 3 (for some 0 n 0 ( δ ) ϵ N such that for n ≥ n 0 ( δ ) there is a duplication among the sums a i + a j + a k ( i ≤ j ≤ k )? Erdos and Turan's question is: if A = ( a i ) i = 1 ∞ is an increasing sequence of natural numbers with a k ≤ ck 2 for some c > 0 and all k , is it true that lim R(n, A) = +∞ (where R ( n , A ) is the number of ways of writing n as a sum of two elements of A )?