Weyl method

in number theory

A method for obtaining non-trivial estimates of trigonometric sums (cf. Trigonometric sum) of the form

$$ \tag{1 } S( f ) = \sum _ {1 \leq x \leq P } e ^ {2 \pi i f( x) } , $$

where

$$ f( x) = \alpha _ {n} x ^ {n} + \dots + \alpha _ {1} x $$

and $ \alpha _ {n} \dots \alpha _ {1} $ are arbitrary real numbers. Developed by H. Weyl [1] to establish criteria for uniform distribution (cf. Weyl criterion).

The method may be described as follows. The sum (1) is raised to the power $ \rho _ {0} = 2 ^ {n-} 1 $ by successive squaring operations in order to reduce the degree of the polynomial $ f( x) $. Thus, in the first stage

$$ S ^ {2} ( f ) = \sum _ {\lambda _ {1} \neq 0 } \sum _ { x } e ^ {2 \pi i ( f( x+ \lambda _ {1} )- f( x)) } + O( P), $$

where the summations are performed over intervals of lengths $ \ll P $; now

$$ f( x+ \lambda _ {1} ) - f( x) = \ n \alpha _ {n} \lambda _ {1} x ^ {n-} 1 + \dots $$

is a polynomial of degree $ n - 1 $ in $ x $( the symbols $ O( P) $, $ \ll P $ denote magnitudes of order $ P $). At the $ ( n - 1) $- st step one obtains inner sum

$$ \tag{2 } S( \alpha )= \sum _ {x= a+ 1 } ^ { {a+ } P _ {1} } e ^ {2 \pi i \alpha x } , $$

where $ P _ {1} \leq P $, $ \alpha = n! \lambda _ {1} {} \dots \lambda _ {n-} 1 \alpha _ {n} $, $ \lambda _ {i} \neq 0 $. Sums of the form (2) are estimated using the inequality

$$ | S( \alpha ) | \leq \min \left ( P _ {1} , \frac{1}{| \sin \pi \alpha | } \right ) , $$

and the resulting estimate is:

$$ \tag{3 } | S( f ) | ^ {\rho _ {0} } \leq P ^ {\rho _ {0} - 1 } + $$

$$ + P ^ {\rho _ {0} {- n + \epsilon } } \sum _ {0 < y \leq P ^ {n-} 1 } \min \left ( P _ {1} , \frac{1}{| \sin \pi y \alpha _ {n} | } \right ) . $$

The inequality (3) yields different estimates for the sum (1) in case $ 1/ | \sin \pi y \alpha _ {n} | $ is small in comparison to $ P $. These estimates depend on the accuracy with which the coefficient $ \alpha _ {n} $ of the polynomial $ f( x) $ can be approximated by rational fractions.

Example. Let

$$ \left | \alpha _ {n} - \frac{p}{q} \right | \leq \frac{1}{q ^ {2} } , \ ( a, q) = 1. $$

Then

$$ | S( f ) | \ll P ^ {1 + \epsilon } q ^ \epsilon \left ( \frac{1}{P} + \frac{1}{q} + \frac{q}{P ^ {n} } \right ) ^ {\rho _ {0} } . $$

In particular, if

$$ P \leq q \leq P ^ {n-} 1 , $$

then

$$ | S( f) | \ll P ^ {1- \rho _ {0} + \epsilon } . $$

Weyl's method gives solutions, to a first approximation, of several important problems in number theory. The estimate (3) and its corollaries were used to study the distribution modulo one of the polynomial $ f( x) $. G.H. Hardy and J.E. Littlewood (1919) gave a solution of the Waring problem which was based on estimating the sums (1) by Weyl's method. They could thus estimate the values of $ r= r( k) $ for which the equation

$$ N = x _ {1} ^ {k} + \dots + x _ {r} ^ {k} , $$

where $ N > 0 $ is an integer and $ x _ {i} $ are integers, is solvable, and even gave an asymptotic formula for the number of solutions. A generalization of the estimate (3) to the case of functions $ f( x) $ which are not polynomials but are in a certain sense close to polynomials, resulted in the improvement of certain theorems in the theory of the distribution of prime numbers (an estimate of the difference between two successive prime numbers and an estimate of the residual term in the asymptotic formula for the number $ \pi ( N) $ of prime numbers not exceeding $ N $).

The insufficient strength of the estimates obtained by Weyl's method is due to the high power $ \rho _ {0} $ to which the sum $ S( f ) $ is raised. J. van der Corput proposed a somewhat improved method for estimating the sums (1). The Vinogradov method yields a very accurate upper bound for the integral

$$ \int\limits _ { 0 } ^ { 1 } \dots \int\limits _ { 0 } ^ { 1 } | S( f ) | ^ {2b} d \alpha _ {n} \dots d \alpha _ {1} $$

already for $ b \geq cn ^ {2} $( $ c > 0 $ a constant, $ n \geq 2 $). This estimate (cf. Vinogradov theorem about the average) may be used to deduce essentially new estimates of Weyl sums (1) (with reduction factor $ P ^ {- \rho } $, $ \rho = c _ {1} n ^ {2} \mathop{\rm ln} n $; $ c _ {1} > 0 $ a constant), which cannot be attained by Weyl's method.

References

[1]	H. Weyl, "Ueber die Gleichverteilung von Zahlen mod. Eins" Math. Ann. , 77 (1916) pp. 313–352
[2]	I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian)