Erdős–Turán

Erdős–Turán
Name	Paul Erdős and Pál Turán (eponym)
Known for	Erdős–Turán conjecture, Erdős–Turán inequality, additive number theory

Erdős–Turán.

The term denotes a cluster of results and conjectures in additive number theory and analytic number theory associated with collaborations between Paul Erdős and Pál Turán. The phrase appears across statements linking asymptotic density, additive bases, trigonometric polynomials, uniform distribution, and probabilistic methods. Their work bridges techniques from combinatorial number theory, harmonic analysis, sieve theory, metric number theory, and complex analysis.

Eponym and Overview

The eponym honors Paul Erdős and Pál Turán for joint and related contributions during the mid‑20th century. Erdős, noted for collaborations with Alfréd Rényi, Ronald Graham, Endre Szemerédi, and Miklós Ajtai, brought probabilistic and combinatorial perspectives; Turán, linked to Pál Turán Prize namesakes and collaborations with Gábor Szegő and András Hajnal, emphasized analytic and extremal approaches. The phrase labels multiple distinct statements: a conjecture on additive bases, an inequality bounding trigonometric sums, plus corollaries used by researchers such as Jean Bourgain, Ben Green, Terry Tao, Imre Z. Ruzsa, and Katalin Győri.

Erdős–Turán Conjecture on Additive Bases

The conjecture asserts that for any basis A of the natural numbers with representation function r_A(n) (counting representations as sums of two elements of A), r_A(n) tends to infinity. Originating in a joint program of Erdős and Turán, it influenced work by Nathanson, Deshouillers, Plagne, Ruzsa, Sándor Z. Kiss, and Miklós Bóna. Partial results connect to the Sidon sets studied by Simon Sidon and to structural theorems by Freiman and Imre Z. Ruzsa. Approaches employ tools developed by Vaughan, Heath-Brown, Iwaniec, and Kloosterman-type analyses, as well as additive combinatorics techniques advanced by Green and Tao.

Erdős–Turán Inequality

The inequality provides a quantitative bound relating discrepancy of distributions on the unit circle to Fourier coefficients of associated measures or trigonometric polynomials. It is used in conjunction with classical results of Weyl, Khinchin, Khintchine, and Vinogradov on uniform distribution and exponential sums. The inequality underpins estimates by Hardy and Littlewood in the study of trigonometric series, and it has been deployed in modern treatments by Montgomery and Iwaniec when analyzing zeros of L-series and exponential sum bounds in sieve theory contexts.

Applications in Additive Number Theory

Erdős–Turán statements influence numerous problems: growth of representation functions in bases (pursued by Nathanson, Sárközy, Deshouillers), inverse problems like Freiman-type structure (pursued by Freiman, Ruzsa, Green), and distribution results for sequences linked to Beatty sequences studied by Rayleigh and Beatty. The inequality is applied in estimating discrepancy in sequences considered by Kuipers and Niederreiter, and in quantitative problems related to the Goldbach conjecture efforts of Vinogradov, Estermann, and Chen Jingrun. Connections extend to combinatorial theorems such as those proven by Erdős with Ginzburg and to probabilistic models developed by Feller and Kolmogorov in stochastic number theory.

History and Development

Initial formulations emerged from correspondence and papers by Erdős and Turán in the 1940s and 1950s during interactions involving John von Neumann‑era mathematicians and centers at Princeton University, University of Szeged, and Eötvös Loránd University. Early dissemination involved seminars where contemporaries such as Paul Halmos, Norbert Wiener, G. H. Hardy, and S. Ramanujan influenced the analytic framing. Subsequent decades saw extensions by Beurling, Bohr, Kronecker, and later refinements using combinatorial frameworks by Szemerédi and algorithmic perspectives by Erdős collaborators including Graham and Székely.

Related Results and Generalizations

Generalizations relate to higher‑order bases, representation by k‑sums studied by Erdős and Graham, probabilistic bases considered by Erdős with Rényi, and local–global principles echoing work by Mordell and Hasse. Comparable inequalities include bounds by Weyl, van der Corput, and Hooley on exponential sums, and extensions by Bourgain on multilinear estimates. Recent advances by Green, Tao, Ben Green, Tao, and Conlon have recast aspects into modern additive combinatorics, while computational searches by groups at University of Chicago, MIT, and University of Cambridge continue to test instances numerically.

Category:Additive number theory