An exponential sum of the form

where is a real polynomial (Weyl 1914, 1916; Montgomery 2001). Writing

a notation introduced by Vinogradov, Weyl observed that

			(3)
			(4)
			(5)
			(6)

a process known as Weyl differencing (Montgomery 2001).

Weyl was able to use this process to show that if

is a real polynomial and at least one of , ..., is irrational, then is uniformly distributed (mod 1).