L1-Norm of Steinhaus chaos on the polydisc

Let J n ⊂ [ 1 , n ] , n = 1 , 2 , … be increasing sets of mutually coprime numbers. Under reasonable conditions on the coefficient sequence { c n j } n , j , we show that lim T → ∞ 1 T ∫ 0 T ∑ j ∈ J n c n j j i t d t ∼ π 2 ∑ j ∈ J n ( c n j ) 2 1 / 2 as n → ∞ . We also show by means of an elementary device that for all 0 < α < 2 , lim T → ∞ 1 T ∫ 0 T ∑ n = 1 N n - i t α d t 1 / α ≥ C α N 1 2 ( log N ) 1 α - 1 2 . the proof uses Ayyad, Cochrane and Zheng estimate on the number of solutions of the equation x 1 x 2 = x 3 x 4 . In the case α = 1 , this approaches Helson’s bound up to a factor ( log N ) 1 / 4 .