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I = lim inf v n-a* log n The purpose of this paper is to combine the methods used in two earlier papers' in order to prove the following theorem. THEOREM. (1) 1 5 c(1 + 4e)/5, where c 1 -c, so that (1) is an improvement on (2) only if 0 is not too close to unity, in fact, if 0

Let f(m;n) denote the largest integer so that, given any m integers a1 < … < am in [1, 2n], one can always choose f integers b1 < … < bf from [1, n], so that bi + bj = a1 (1 ≤ i ≤ j ≤ f; l ≤ l ≤ m) will never hold. Trivially f(m; n) ≥ n/ (m + 1). In this paper we shall attempt to improve upon this trivial bound by exploiting the possible irregularities of distribution of the sequence among certain congruence classes. One of our main results isprovided m ≥ log n. Related questions and results are also discussed.

A one parameter solution of th equation was given by Ryley in 1825. Others were found by Richmond and myself. In the Landau memorial volume [1] recently published, there appears a joint paper, with the above title, of Davenport and Landau dating from 1935. They prove that if n is a positive integer, then positive rational solutions of (1) exist with denominators of order of magnitude O ( n 2 ). Their proof depended upon a two parameter solution of (1) due to Richmond and is very complicated.