On the sum of sine products

Given N 1 and d 1, we show the existence of positive integers λkj for 1 k N and 1 j d such that N k=1 d j=1 sin λkj xj CN(d+1)/(2d+1) for all real x1, . . ., xN, where C is a constant that depends only on d. This extends a result of Bourgain for the case d = 1.  2003 Elsevier Inc. All rights reserved