An exponential sum of the form

(1)

where  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).

REFERENCES:

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. Providence, 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. Berlin: 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. Berlin: Springer-Verlag, pp. 563-599, 1968. Also reprinted in Selecta Hermann Weyl. Basel, Switzerland: Birkhäuser, pp. 111-147, 1956.