Littlewood [5, Problem 4.19, originally 4] conjectured that there is an absolute constant C > 0 such that, for every sequence of distinct integers n1, n2, n3, …, if
Cohen  showed
for some absolute constant C, with b = 1/8. Davenport  gave a more explicit version of Cohen's proof and improved the estimate to b = 1/4. Pichorides  added another refinement to obtain b = ½, and has, more recently, obtained ‖fN‖1>C(log N)1/2. This seems to be the best estimate so far without restriction on the sequences. We shall show that the methods of Davenport and Pichorides may be extended to obtain better results for certain classes of sequences. Specifically, we prove the following theorems.