An exponential sum of the form
 |
(1) |
where P(n) is a real polynomial (Weyl 1914, 1916; Montgomery 2001). Writing
 |
(2) |
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
 |
(7) |
is a real polynomial and at least one of 
,
is irrational, then 
is uniformly distributed (mod 1).
van der Corput's Inequality, Weyl's Criterion
Berry, M. V. and Goldberg, J. "Renormalisation of Curlicues." Nonlinearity 61, 1-26, 1988.
Lehmer, D. H. and Lehmer, E. "Picturesque Exponential Sums, I." Amer. Math. Monthly 86, 725-733, 1979.
Montgomery, H. L. "Harmonic Analysis as Found in Analytic Number Theory." In Twentieth Century Harmonic Analysis--A Celebration. Proceedings of the NATO Advanced Study Institute Held in Il Ciocco, July 2-15, 2000 (Ed. J. S. Byrnes). Dordrecht, Netherlands: Kluwer, pp. 271-293, 2001.
Montgomery, H. L. Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis.rovidence, RI: Amer. Math. Soc., 1994.
Pickover, C. A. "Is the Fractal Golden Curlicue Cold?" Visual Comput. 11, 309-312, 1995.
Stewart, I. Another Fine Math You've Got Me Into.... New York: Freeman, 1992.
Weyl, H. "Über ein Problem aus dem Gebiete der diophantischen Approximationen." Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., 234-244, 1914. Reprinted in Gesammelte Abhandlungen, Band I. Berline: Springer-Verlag, pp. 487-497, 1968.
Weyl, H. "Über die Gleichverteilung von Zahlen mod. Eins." Math. Ann. 77, 313-352, 1916. Reprinted in Gesammelte Abhandlungen, Band I. Berline: Springer-Verlag, pp. 563-599, 1968. Also reprinted in Selecta Hermann Weyl. Basel, Switzerland: Birkhäuser, pp. 111-147, 1956.