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

$|c-a/q|\le t{q}^{-2},$

for some t greater than or equal to 1, then for any positive real number $\epsilon$ one has

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

This inequality will only be useful when

$q<N^k,$

for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as $\le N$ 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. http://link.springer.com/10.1023/A:1013873301435.