We consider higher dimensional versions of discrete fractional integral operators first investigated by Stein and Wainger. Specifically, we define operators Iλ and Jλ for 0 < λ < 1

by

Iλf(n) = X

m∈Z

k

+

f(n − |m|

2

)

|m|

kλ , Jλf(n, t) = X

m∈Z

k

+

f(n − m, t − |m|

2

)

|m|

kλ ;

here Iλ acts on functions of Z, Jλ on functions of Z

k+1. Our interest is in proving the

boundedness of these operators from ℓ

p

to ℓ

q

for appropriate p, q, λ. We show how this

can be done using complex interpolation and ideas originating from the circle method in

number theory; furthermore we consider the case where |m|

2

is replaced by a more general

quadratic form.