Turán's power sum problem

Turán's power sum problem
Name	Turán's power sum problem
Field	Number theory, Combinatorics, Analytic number theory
Introduced	1950s
Person	Paul Turán

Turán's power sum problem is a question in analytic number theory and combinatorics concerning lower bounds for maxima of absolute values of finite power sums of complex numbers on the unit circle. It asks how small the maximum modulus of a trigonometric or algebraic power sum can be given constraints on the coefficients and the number of terms, and it sits at the intersection of problems studied by Paul Erdős, Gábor Szegő, von Neumann, Gauss, and Schur.

Statement of the problem

Let n be a positive integer and let z_1,...,z_n be complex numbers with |z_j| = 1. Turán's power sum problem asks for sharp lower bounds for max_{1 ≤ k ≤ m} |S_k| where S_k = sum_{j=1}^n z_j^k, for specified m (often m = n or m ≥ n). The formulation appears in variants requiring z_j to be distinct, to avoid roots of unity degeneracies, or to impose algebraic integer restrictions studied by Ostrowski and Schur. Typical statements seek explicit functions f(n,m) such that max_{1 ≤ k ≤ m} |S_k| ≥ f(n,m), with equality or near-equality realized by configurations related to roots of unity or to extremal sets considered by Erdős and Laczkovich.

Historical background and motivation

The problem traces to work of Paul Turán in mid-20th century, motivated by questions in Paul Erdős's additive problems and by extremal phenomena studied by Hardy and Littlewood in trigonometric series. Turán's investigations connected to inequalities of Issai Schur and moments studied by Chebyshev and Riemann; later attention came from Bohr-type almost periodicity questions and from spectral distribution problems examined by von Neumann and Wiener. Influential contributions by Paul Erdős, Rényi, Komlós, Szemerédi, and Sárközy linked the problem to equidistribution, discrepancy, and to extremal constructions in combinatorics.

Main results and theorems

Classical lower bounds by Paul Turán show that for m = n one has max_{1 ≤ k ≤ n} |S_k| ≥ c sqrt{n} for an absolute c, with refinements by Paul Erdős and Rényi giving constants and probabilistic constructions attaining comparable sizes. Results by Odlyzko and Granville connected extremal configurations to roots of unity and to cyclotomic patterns studied by Galois. Precise asymptotics for variants where coefficients are ±1 were established by Håstad-style discrepancy analyses and by methods of Mahler and Littlewood for polynomials on the unit circle. Lower bounds employing potential theory trace back to techniques used by Bernstein and Lavrentiev, while upper bounds via explicit constructions use classical sets of roots of unity exploited by Gauss and later by Fejes Tóth.

Methods and techniques

Approaches combine analytic, algebraic, and combinatorial tools: Fourier-analytic methods akin to those of Weil and Wiener; probabilistic methods inspired by Paul Erdős and Rényi; algebraic number theory invoking Kronecker and cyclotomic theory associated with Galois; and potential-theoretic methods tracing to Gauss and Carleman. Extremal combinatorics techniques from Turán-type problems and from Erdős-Szemerédi frameworks are used to build constructions, while matrix-analytic and spectral methods link to work of von Neumann and Schur on eigenvalue distributions. Saddle-point estimates and large deviations from Kolmogorov-style probability theory provide probabilistic concentration used in nonconstructive bounds.

Connections and applications

Turán's power sum problem connects to equidistribution results of Weyl and to discrepancy theory advanced by Jarník-style investigations, with implications for pseudorandomness in constructions studied by Kolmogorov and Rényi. Links to minimal energy point configurations relate to potential-theory work of Kelvin and Thomson, and to polynomial inequalities in approximation theory by Chebyshev and Bernstein. Applications appear in signal processing traditions stemming from Wiener and in spectral analysis of time series considered by von Neumann and Kolmogorov, and in constructions for coding theory influenced by Hamming and Shannon.

Open problems and conjectures

Sharp constants and exact asymptotics remain open in many regimes: determining the optimal f(n,m) for general m relative to n is unresolved, as is the classification of extremal configurations beyond roots of unity patterns studied by Galois and Gauss. Conjectures inspired by Paul Erdős and Paul Turán propose specific growth rates and structural rigidity analogous to rigidity theorems in Szemerédi-type combinatorics. Further links to uniform distribution conjectures of Weyl-type and to spectral gap phenomena studied by Sinai remain active directions for research.

Category:Analytic number theory