Weyl's inequality (number theory)

In number theory, Weyl's inequality, named for Hermann Weyl, states that if M, N, a and q are integers, with a and q coprime, q > 0, and f is a real polynomial of degree k whose leading coefficient c satisfies

${\displaystyle |c-a/q|\leq tq^{-2},}$

for some t greater than or equal to 1, then for any positive real number ${\displaystyle \scriptstyle \varepsilon }$ one has

${\displaystyle \sum _{x=M}^{M+N}\exp(2\pi if(x))=O\left(N^{1+\varepsilon }\left({t \over q}+{1 \over N}+{t \over N^{k-1}}+{q \over N^{k}}\right)^{2^{1-k}}\right){\text{ as }}N\to \infty .}$

This inequality will only be useful when

${\displaystyle q<N^{k},}$

for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as ${\displaystyle \scriptstyle \leq \,N}$ provides a better bound.