Bounds for discrete multilinear spherical maximal functions

We define a discrete version of the bilinear spherical maximal function, and show bilinear l p ( Z d ) × l q ( Z d ) → l r ( Z d ) bounds for d ≥ 3 , 1 p + 1 q ≥ 1 r , r > d d - 2 and p , q ≥ 1 . Due to interpolation, the key estimate is an l p ( Z d ) × l ∞ ( Z d ) → l p ( Z d ) bound, which holds when d ≥ 3 , p > d d - 2 . A key feature of our argument is the use of the circle method which allows us to decouple the dimension from the number of functions compared to the work of Cook.