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
| c − a / q | ≤ t q − 2 , {\displaystyle |c-a/q|\leq tq^{-2},}
for some t greater than or equal to 1, then for any positive real number ε {\displaystyle \scriptstyle \varepsilon } one has
∑ x = M M + N exp ( 2 π i f ( x ) ) = O ( N 1 + ε ( t q + 1 N + t N k − 1 + q N k ) 2 1 − k ) as N → ∞ . {\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
q < N k , {\displaystyle q<N^{k},}
for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as ≤ N {\displaystyle \scriptstyle \leq \,N} provides a better bound.
References Vinogradov, Ivan Matveevich (1954). The method of trigonometrical sums in the theory of numbers. Translated, revised and annotated by K. F. Roth and Anne Davenport, New York: Interscience Publishers Inc. X, 180 p. Allakov, I. A. (2002). "On One Estimate by Weyl and Vinogradov". Siberian Mathematical Journal. 43 (1): 1–4. doi:10.1023/A:1013873301435. S2CID 117556877.