Some remarks on Bh[g] sequences

AbstractAn increasing sequence of natural numbers A = (ai) (finite or infinite) is called Bh[g] if every n ϵ N can be written in at most g ways as a sum of h (h ≥ 2) elements of A. Fh(n,g) is the cardinality of the largest Bh[g] sequence in {1,…,n}. A well-known question asks for bounds on Fh(n, g) (see H. Halberstam and K. Roth, Sequences, Oxford Univ. Press, London/New York, 1966). We obtain bounds on Fh(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 Fh(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. Erdös and P. Turan [J. London Math. Soc. 16 (1941), 212–215]. Bose and Chowla's question is: given natural numbers a1 < a2 … < an < δn3 (for some 0 < δ < 1) does there exist a n0(δ) ϵ N such that for n ≥ n0(δ) there is a duplication among the sums ai + aj + ak (i ≤ j ≤ k)? Erdös and Turan's question is: if A = (ai)i= 1∞ is an increasing sequence of natural numbers with ak ≤ ck2 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)