Bounds for discrete multilinear spherical maximal functions

We define a discrete version of the bilinear spherical maximal function, and show bilinear $$l^{p}(\mathbb {Z}^d)\times l^{q}(\mathbb {Z}^d) \rightarrow l^{r}(\mathbb {Z}^d)$$ bounds for $$d \ge 3$$ , $$\frac{1}{p} + \frac{1}{q} \ge \frac{1}{r}$$ , $$r>\frac{d}{d-2}$$ and $$p,q\ge 1$$ . Due to interpolation, the key estimate is an $$l^{p}(\mathbb {Z}^d)\times l^{\infty }(\mathbb {Z}^d) \rightarrow l^{p}(\mathbb {Z}^d)$$ bound, which holds when $$d \ge 3$$ , $$p>\frac{d}{d-2}$$ . 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.