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| Weyl sum | |
|---|---|
| Name | Weyl sum |
| Field | Analytic number theory |
| Introduced | 1916 |
| Introduced by | Hermann Weyl |
| Main concepts | Exponential sum, equidistribution, Weyl differencing |
| Notable results | Weyl's inequality, Weyl's criterion, van der Corput method |
Weyl sum
A Weyl sum is a finite exponential sum used in analytic number theory, typically of the form sum_{n=1}^N e( f(n) ) with e(x)=exp(2πi x), studied to understand cancellation and distribution phenomena in sequences arising from polynomials and other functions. Such sums connect central problems and figures across number theory, harmonic analysis, and dynamical systems, influencing work by Hermann Weyl, Gábor Szegő, I. M. Vinogradov, John von Neumann, and Paul Erdős. They serve as a bridge between explicit estimates like Weyl's inequality and qualitative criteria such as Weyl's criterion for uniform distribution modulo one.
A Weyl sum commonly denotes S(N; P) = sum_{n=1}^N e( P(n) ) for a polynomial P(x) with real coefficients; variants allow rational functions, smooth phases, or multidimensional inputs. Core properties include periodicity under integer shifts of coefficients, trivial L^2 bounds via orthogonality used by Norbert Wiener and Salomon Bochner, and relations to Fourier analysis in the spirit of Andrey Kolmogorov and Littlewood–Paley theory. For polynomial phases of degree d the classical observation of Hermann Weyl links smallness of S(N; P) to diophantine approximation of leading coefficients, echoing methods developed by Khinchin, Aaronson, and Duffin–Schaeffer style work. Weyl sums obey basic transformations like completion, partial summation comparable to techniques used by J. E. Littlewood and G. H. Hardy in the context of the Hardy–Littlewood circle method.
The systematic study originates in Weyl's 1916 work on uniform distribution, where he introduced both the exponential sum framework and a criterion characterizing equidistribution of sequences modulo one. Weyl's criterion states that a sequence (x_n) is equidistributed mod 1 iff all nonzero integer frequency sums N^{-1} sum_{n=1}^N e(k x_n) tend to zero, a principle later connected to spectral analysis by John von Neumann and to ergodic theory via George Birkhoff and John Hopf. Subsequent decades saw influential contributions from I. M. Vinogradov on trigonometrical sums, G. H. Hardy and J. E. Littlewood on mean-value estimates, and from Harald Bohr on almost periodicity. Developments by Paul Erdős and A. Selberg extended diophantine approximation links, while modern refinements owe to work by Enrico Bombieri, Hugh Montgomery, Roger Heath-Brown, and Terence Tao.
Weyl's inequality gives a power-saving bound for polynomial Weyl sums by relating S(N; P) to rational approximations of leading coefficients; its proof uses successive differencing and combinatorial identities reminiscent of techniques by John von Neumann and I. M. Vinogradov. Complementary is van der Corput's method of exponential sum estimation, with A-process and B-process variants developed by J. G. van der Corput and refined by D. R. Heath-Brown and Enrico Bombieri. Modern incarnations include Weyl differencing leading to exponent pairs, and bounds from mean value theorems associated to Robert Vaughan and Jean Bourgain. Breakthroughs such as the efficient congruencing method of T. D. Wooley and decoupling approaches by Jean Bourgain, Colin Demeter, and Larry Guth have produced stronger estimates for higher degree phases and critical mean values.
Weyl sums underpin many results in analytic number theory: distribution of polynomial sequences modulo one, estimates for error terms in counting lattice points related to Gauss circle problem, and bounds for classical objects like the Riemann zeta function via exponential sum inputs used by G. H. Hardy and Alan Turing. They play a role in the Hardy–Littlewood circle method for additive problems including Waring's problem solved by Ivan Vinogradov and refined by R. C. Vaughan and T. D. Wooley. Equidistribution results in homogeneous dynamics link to works by Marina Ratner, Grigory Margulis, and Elon Lindenstrauss, while applications to pseudorandomness and discrepancy feature in research by Paul Erdős, József Beck, and William M. Schmidt.
Generalised Weyl sums consider sums over integer vectors n in Z^s with phases given by multivariate polynomials or forms; these are central to Diophantine approximation and to counting integer solutions of systems of polynomial equations studied by H. Davenport, Roger Heath-Brown, and Timothy Browning. Multidimensional decoupling theorems by Jean Bourgain and Colin Demeter yield new bounds for such sums, with implications for restriction theory associated to Elias Stein and Terence Tao. Further generalisations include exponential integrals over manifolds encountered in work of Erik Christopher Zeeman and oscillatory integral estimates linked to Christopher Sogge.
Proofs employ harmonic analysis, diophantine approximation, and combinatorial differencing. Weyl differencing reduces degree, while van der Corput's processes produce exponent pairs (k, l) used systematically to propagate bounds; exponent pair technology originates from work by G. H. Hardy, J. E. Littlewood, and later formalised by A. A. Karatsuba and Y. V. Linnik. Contemporary methods incorporate decoupling, efficient congruencing, and multilinear harmonic analysis pioneered by Bennett–Carbery–Tao and advanced by Jean Bourgain, Larry Guth, and Ciprian Demeter. These approaches link to spectral methods in quantum chaos studied by Zelditch and to ergodic techniques from Furstenberg.